A Topological Characterization of Modulo- p Arguments and Implications for Necklace Splitting
Aris Filos-Ratsikas, Alexandros Hollender, Katerina Sotiraki, Manolis Zampetakis
AA Topological Characterization of Modulo- p Arguments andImplications for Necklace Splitting
Aris Filos-Ratsikas Alexandros Hollender
University of Liverpool, United Kingdom University of Oxford, United Kingdom
[email protected] [email protected]
Katerina Sotiraki Manolis Zampetakis
University of California Berkeley, USA University of California Berkeley, USA [email protected] [email protected]
Abstract
The classes PPA- p have attracted attention lately, because they are the main candidatesfor capturing the complexity of Necklace Splitting with p thieves , for prime p . However, theseclasses were not known to have complete problems of a topological nature, which impedes anyprogress towards settling the complexity of the Necklace Splitting problem. On the contrary,topological problems have been pivotal in obtaining completeness results for PPAD and PPA,such as the PPAD-completeness of finding a Nash equilibrium [Daskalakis et al., 2009, Chenet al., 2009b] and the PPA-completeness of Necklace Splitting with 2 thieves [Filos-Ratsikas andGoldberg, 2019].In this paper, we provide the first topological characterization of the classes PPA- p . First,we show that the computational problem associated with a simple generalization of Tucker’sLemma, termed p - polygon -T ucker , as well as the associated Borsuk-Ulam-type theorem, p - polygon -B orsuk -U lam , are PPA- p -complete. Then, we show that the computational versionof the well-known BSS Theorem [Bárány, Shlosman, and Szücs, 1981], as well as the associatedBSS-T ucker problem are PPA- p -complete. Finally, using a different generalization of Tucker’sLemma (termed Z p - star -T ucker ), which we prove to be PPA- p -complete, we prove that p -thiefNecklace Splitting is in PPA- p . This latter result gives a new combinatorial proof for theNecklace Splitting theorem, the only proof of this nature other than that of Meunier [2014].All of our containment results are obtained through a new combinatorial proof for Z p -versions of Tucker’s lemma that is a natural generalization of the standard combinatorial proofof Tucker’s lemma by Freund and Todd [1981]. We believe that this new proof technique is ofindependent interest. a r X i v : . [ c s . CC ] J a n ontents k -Polygon Borsuk-Ulam Theorem . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 Complexity of Finding Solutions to the BSS Theorem . . . . . . . . . . . . . . 71.1.3 Necklace Splitting with p thieves is in PPA- p . . . . . . . . . . . . . . . . . . . 91.2 Discussion and Further Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 k -Polygon Borsuk-Ulam: a PPA- k [ ] -complete Problem in 2D-space 14 k -Polygon Tucker’s Lemma and k -Polygon Borsuk-Ulam in PPA- k [ ] . . . . . . . . 153.1.1 Equivalence of k -Polygon Tucker and k -Polygon Borsuk-Ulam . . . . . . . . 163.1.2 Proof of k -Polygon Tucker’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . 173.1.3 Computational Problems and Containment in PPA- k [ ] . . . . . . . . . . . . 223.2 k -Polygon Borsuk-Ulam and k -Polygon Tucker are PPA- k [ ] -hard . . . . . . . . . . 27 p -complete 33 ucker Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.3 Computational problems: p -BSS-T ucker and p -BSS . . . . . . . . . . . . . . . . . . . 364.4 BSS-T ucker is PPA- p -complete . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Z p - star -T ucker Lemma: Statement and PPA- p -completeness 40 Z p - star -T ucker is in PPA- p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415.1.1 Proof Overview of Theorem 5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 425.2 Z p - star -T ucker is PPA- p -hard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 p Thieves lies in PPA- p onsensus -1/ p -D ivision to Z p - star -T ucker . . . . . . . . . . . . . 48 A.1 Simplicial Complexes, Value & Index Functions and Triangulations . . . . . . . . . . 55A.2 The Borsuk-Ulam Theorem and Tucker’s Lemma . . . . . . . . . . . . . . . . . . . . 57A.3 Z p -equivariant tie-breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 B ( p , n ) -BSS-Tucker reduces to ( p , n + ) -BSS-Tucker 59C Z p - star -T ucker is in PPA- p , Full Proof 59 Introduction
The class TFNP [Megiddo and Papadimitriou, 1991] is the class of
Total Search Problems in NP , i.e.,problems for which a solution is always guaranteed to exist, and can be verified in polynomialtime. In a seminal paper, attempting to capture the complexity of numerous interesting problems,Papadimitriou [1994] defined several subclasses of TFNP, such as PPAD, PPA and PPP, amongothers, each of which is associated with a different existence principle. For example, PPAD isbased on the principle that given a source in a directed graph with in-degree and out-degree atmost 1, there must exist another vertex of degree 1; PPA is based on a similar principle on anundirected graph, and PPP is based on the pigeonhole principle.Among those, PPAD has been largely successful in capturing the complexity of many well-known problems, most prominently that of computing Nash equilibria in games [Daskalakis et al.,2009, Chen et al., 2009b]. PPA and PPP have been more elusive in that regard for nearly twodecades, until the recent results of Filos-Ratsikas and Goldberg [2018, 2019] and Sotiraki et al.[2018], who provided the first “natural” complete problems for these classes respectively. Here, a“natural” problem is one that does not explicitly include a polynomial-sized circuit in its definition(as termed in [Grigni, 2001]). Interestingly, the PPA-complete problems in [Filos-Ratsikas andGoldberg, 2018, 2019] that solidified the status of PPA as a class containing such natural problemswere the well-known
Necklace Splitting problem for two thieves as well as its continuous variant(coined the
Consensus-Halving problem in [Simmons and Su, 2003]).The Necklace Splitting problem is a classical problem in combinatorics, dating back to themid-1980s and the works of Goldberg and West [1985], Alon and West [1986] and Alon [1987],among others. In this problem, k thieves are aiming to split an open necklace containing n beadsof t different colors (with exactly k · a i beads of color i , for some a i ∈ N ), such that each thiefreceives exactly a i beads of color i . Furthermore, the thieves are allowed to use only ( k − ) t cutsto obtain this division. A solution to the Necklace Splitting problem is guaranteed to exist, aswas proven by Alon [1987]. Earlier on, Goldberg and West [1985] and Alon and West [1986] hadproven the existence of a solution for the case of 2 thieves; this result invokes a fundamental toolfrom mathematics, the Borsuk-Ulam Theorem [Borsuk, 1933]. Alon’s proof for the general caseproceeds in two steps. First, using a simple argument, he proves that if the theorem holds forany prime number p of thieves, it also holds for any other number of thieves. In the second step,which is significantly more involved, he proves the theorem for any prime number by using the BSS Theorem , a generalization of the Borsuk-Ulam Theorem due to Bárány, Shlosman, and Szücs[1981].Questions about the computational complexity of finding a solution were raised explicitly asearly as when the first existence results were proven [Goldberg and West, 1985] and then lateron by a series of papers [Alon, 1988, 1990, Meunier, 2008, Meunier and Seb˝o, 2009, Meunier andNeveu, 2012]. The first definitive answer was provided by Filos-Ratsikas and Goldberg [2019], viatheir PPA-completeness result, following an initial PPAD-hardness result by Filos-Ratsikas et al.[2018]. Crucially however, their result only applies to the case of two thieves. In fact, the authorsobserved (also referencing de Longueville and Živaljevi´c [2006] and Meunier [2014] as previouslyhaving made similar observations) that the version of the problem with p ≥ p -thief Necklace Splitting is complete for the computational class PPA- p , also defined by The authors also extend the PPA membership straightforwardly to numbers of thieves which are powers of 2. p in a bipartite graph, find another such vertex. It follows fromthe definition that PPA-2 = PPA.The classes PPA- p have been very recently studied by Hollender [2019] and Göös et al. [2020].The authors of the latter paper in fact provide the first PPA- p -completeness result for a naturalproblem, a computational version of the Chevalley-Warning theorem. Importantly, they were ableto obtain their completeness result via reductions to and from equivalent variants of the canonicalproblems of the class. To prove any results about p -thief Necklace Splitting however, such anapproach seems insufficient.To see this, note that the results for p =
2, both for the PPA-membership [Filos-Ratsikas et al.,2018] and for PPA-hardness [Filos-Ratsikas and Goldberg, 2019] of the problem, are obtained viareductions to/from a computational version of
Tucker’s Lemma [Tucker, 1945], a discrete analogueto the Borsuk-Ulam theorem, proven to be PPA-complete by Papadimitriou [1994] and Aisenberget al. [2020]. Tucker’s lemma asserts that if we have an antipodally symmetric triangulation of a d -dimensional ball B and a labeling function which assigns complementary labels to antipodalpoints on the boundary, then there is a complementary edge , i.e., two adjacent points with equal-and-opposite labels. This connection is not a coincidence; for example, the idea in [Filos-Ratsikaset al., 2018] is in fact a direct adaptation of a combinatorial existence proof of Simmons and Su[2003] for the Consensus-Halving problem, which goes via Tucker’s lemma.In order to obtain a PPA- p result for p -thief Necklace Splitting, it seems rather imperative todevelop an “arsenal” of computational problems of a related nature, that we could reduce to/from,namely generalizations of the Borsuk-Ulam theorem and Tucker’s lemma. Such “topologicalcharacterizations” did not only enable researchers in settling the complexity of the problem (andsome other related problems) for PPA = PPA-2, but also facilitated the success of PPAD to theutmost extent, seeing as virtually all the related important results go via the computational versionsof the associated topological theorems (specifically
Brouwer’s Fixed-Point Theorem [Brouwer, 1911]and
Sperner’s Lemma [Sperner, 1928]). In a very related manner, Simmons and Su [2003] stressedthe importance of obtaining such a generalization, in the quest for obtaining a combinatorial proofof
Consensus- k-Division (the generalization of Consensus-Halving) and therefore, for NecklaceSplitting with k thieves, for k >
2. Lastly, Göös et al. [2020] explicitly raised the complexity of oneof these generalizations, the BSS Theorem, as an open problem.
In this paper, we obtain such a topological characterization of the classes PPA- p , for all primes p ≥
3. Namely, we provide generalizations of the computational versions of the Borsuk-Ulamtheorem and Tucker’s lemma, parameterized by p , which are complete for PPA- p . A highlight ofour generalizations is the PPA- p completeness of the computational version of the BSS Theorem.Finally, we use a further generalization to prove that Necklace Splitting with p thieves lies in PPA- p .The strength of our results lies in that (cid:5) they solidify the status of the classes PPA- p as classes containing interesting well-knownproblems (adding to the recent results of Göös et al. [2020]), and To be more precise, several important PPAD-hardness results were obtained via reductions from the
GeneralizedCircuit problem [Daskalakis et al., 2009, Chen et al., 2009b, Rubinstein, 2018], which was however proven to bePPAD-complete via the aforementioned topological problems. they set up an essential toolkit for obtaining more completeness results for the classes, e.g.,possibly the PPA- p -completeness of p -thief Necklace Splitting.All of our PPA- p -membership results are obtained via a new combinatorial proof for Z p -versionsof Tucker’s lemma. This new proof can be seen as a natural generalization of the standardcombinatorial proof of Tucker’s lemma by Freund and Todd [1981]. Thus, as a byproduct of ourtechniques we also obtain the following results:1. A combinatorial proof of the BSS theorem. The original proof by Bárány et al. [1981] is notcombinatorial, as it uses various tools from algebraic topology. Using our new technique,we are able to provide the first combinatorial proof for this theorem.2. A combinatorial proof of the Necklace Splitting theorem. The existence of such a combinato-rial proof had been an open problem since [Alon, 1987]. This open problem was solved byMeunier [2014] using a rather complicated argument. In contrast, our new combinatorialproof uses more elementary tools and is conceptually simpler than Meunier’s proof.3. A stronger statement of the continuous Necklace Splitting theorem which is called Consensus-1/ p -Division [Alon, 1987, Simmons and Su, 2003]. The main advantage of our new theoremis that it actually works for valuation functions that are not necessarily additive and non-negative, for details see Theorem 6.5.As a result, we believe that this new technique is of independent interest and will be useful forproviding combinatorial proofs of other topological existence theorems such as Dold’s TheoremDold [1983]. We remark here that although the original proof of the Necklace Splitting theorem in[Alon, 1987] is via the BSS Theorem, our PPA- p membership result for Necklace Splitting does not go via the PPA- p membership result that we prove for BSS. This is due to the fact that for severalsteps of the proof of Alon [1987], it is quite unclear whether they can be carried out in polynomialtime. Instead, we construct a reduction directly from Z p - star -T ucker , which we prove to bePPA- p -complete.In the remainder of this section, we informally state our main problems and results, and we give ashort and high-level description of our proof techniques. We start with a topological theorem thatis easy to state in Section 1.1.1, which we call k -Polygon Tucker’s Lemma. Then, we present ourresults about the completeness of the well-studied BSS Theorem in Section 1.1.2. Finally, we brieflyexplain how we get our new combinatorial proof of Necklace Splitting and the containment inPPA- p , in Section 1.1.3.Throughout this paper, unless otherwise specified, k denotes an integer larger or equal to 2, and p denotes a prime number. k -Polygon Borsuk-Ulam Theorem The k -Polygon Borsuk-Ulam theorem can be understood via the corresponding statement ofBorsuk-Ulam in 2 dimensions. First, let us introduce the following notion of equivariance for afunction. Let S be the unit circle in 2-dimensions and B be the unit disk. We say that a function g : S → R is equivariant to a rotation of a ◦ degrees if whenever we rotate the input x by a ◦ , theimage g ( x ) is also rotated by a ◦ degrees. We extend this definition to functions f : B → R andwe say that f is equivariant to a rotation of a ◦ degrees on the boundary if the restriction of f to theboundary S is equivariant to a rotation of a ◦ degrees. Using this language, the following is one5f the many equivalent ways to state the classical Borsuk-Ulam Theorem in 2 dimensions (see thebook by Matoušek [2008] for various equivalent versions). Informal Theorem 1 (2D B orsuk -U lam T heorem ) . Let f : B → R be a continuous function that isequivariant to a rotation of ◦ degrees on the boundary. Then, there exists x (cid:63) ∈ B such that f ( x (cid:63) ) = . The generalization, that we call k -Polygon Borsuk-Ulam Theorem, comes as a clean extension ofBorsuk-Ulam where instead of assuming equivariance to a rotation of 180 ◦ degrees, we assumeequivariance to a rotation of ( k ) ◦ degrees. Informal Theorem 2 ( k -P olygon B orsuk -U lam T heorem ) . Let f : B → R be a continuousfunction that is equivariant to a rotation of ◦ k degrees on the boundary. Then, there exists x (cid:63) ∈ B suchthat f ( x (cid:63) ) = . Apart from its own mathematical interest, the k -Polygon Borsuk-Ulam Theorem is essential for ourresults, since it serves as a stepping stone towards showing all our topological PPA- p -completenessresults. Also, it is the only one of our topological problems which is defined for any k ≥
2, andthus the only problem which we are able to relate to the classes PPA- k , for general k .To prove the k -Polygon Borsuk-Ulam Theorem, we deviate from the topological techniquesthat have been used in the proof of similar extensions of the Borsuk-Ulam Theorem [Bárány et al.,1981] and we instead provide a combinatorial proof of a corresponding generalization of Tucker’slemma. We call this lemma k -Polygon Tucker’s Lemma and an informal statement follows. Informal Theorem 3 ( k -P olygon T ucker ’ s L emma ) . For k ≥ , let T be a triangulation of B withk-fold rotationally symmetric boundary. Suppose that every vertex x ∈ T has a color λ ( x ) ∈ Z k such that λ is equivariant to a rotation of ◦ k degrees on the boundary. Then, in T there exists (i) a trichromatictriangle, or (ii) an edge with distinct non-consecutive colors. The coloring function λ is equivariant to a rotation of α ◦ degrees on the boundary if whenever theargument x is rotated by α ◦ , the color is increased by 1 ( mod k ) . Arguably, for k = k -Polygon Tucker’s Lemma bears more resemblance to Sperner’s Lemma than to theoriginal version of Tucker’s Lemma, because the solution is necessarily a trichromatic triangle.However, this is due to the fact that we are considering only the two-dimensional case here. Theconnection with the original version of Tucker’s Lemma will become more apparent when wepresent other high-dimensional modulo- p generalizations of Tucker’s Lemma.In the proof of k -Polygon Tucker’s Lemma, the only inefficient step is the use of a modulo-k counting argument. A simple way to visualize this argument is to imagine that if a space hascardinality that is non-zero modulo k and if we can group points in this space that are non-solutions into groups of size k , then in this space there should exist a solution. This kind ofexistential argument has been formalized by Papadimitriou in his seminal paper [Papadimitriou,1994] and there have been various instantiations of this principle from which we can definecorresponding computational problems [Göös et al., 2020, Hollender, 2019]. In this paper, wemostly rely on the following instantiation defined by Hollender [2019]:I mbalance - mod - k : Given a directed graph and a vertex that is imbalanced-mod-k , i.e., ( out - degree ) − ( in - degree ) (cid:54) = ( mod k ) , find another such vertex.Our main technical contribution in this section is to show that the computational problemassociated with k -Polygon Tucker’s Lemma is polynomial time equivalent to the computational6ersion of I mbalance - mod - k . Towards this goal, we provide a generalization of the standard path-following proof of Tucker’s lemma by Freund and Todd [1981]. Importantly, in our proof (which isa reduction to I mbalance - mod - k ) the edges of the path are directed by using a consistent directionof triangulations (inspired by the idea of Freund [1984]). This technique was in fact incorrectlyapplied in the past to the case of k =
2, leading to a false statement of PPAD-membership forBorsuk-Ulam. However, it turns out that the technique is very relevant in showing the equivalencebetween topological problems and modulo- k arguments for k >
2. We illustrate the appropriateway to use this technique via the I mbalance - mod - k problem and we believe that this will beuseful for future reductions in PPA- k .The computational equivalence between k -Polygon Tucker’s Lemma and I mbalance - mod - k together with the computational equivalence of k -Polygon Tucker’s Lemma and the k -PolygonBorsuk-Ulam implies the following theorem, which is our main result in this section. Informal Theorem 4.
If k is a prime power, the computational problem associated with the k-PolygonBorsuk-Ulam Theorem is
PPA- k-complete. If k is not a prime power, then it is complete for a subclass of
PPA- k, denoted by
PPA- k [ ] . The reason for this differentiation depending on whether k is a prime power or not, is that weactually show equivalence of k -Polygon Tucker’s Lemma with a special case of I mbalance - mod - k .If k is a prime power, then this special case is in fact PPA- k -complete, but in general it can beweaker.The formal definitions and the complete proofs, including the proof of Informal Theorem 4,about the k -Polygon Borsuk-Ulam Theorem appear in Section 3. In this section, we present our results regarding the computational complexity of finding solutionsguaranteed to exist by the BSS Theorem, a famous generalization of the celebrated Borsuk-UlamTheorem. The BSS Theorem should be thought of as the corresponding k -Polygon Borsuk-Ulamin higher dimensions. We clarify though that the BSS Theorem only works with k = p where p isa prime, and it applies only when the number of dimensions is a multiple of p −
1. Hence, for p ≥ p -Polygon Borsuk-Ulam Theorem does not follow directly from the BSS Theorem. Thisis why we consider it a separate result and devote a separate section to it.When moving to more than two dimensions, we need to find an equivariance notion corre-sponding to the equivariance of a rotation that we defined in the previous section. A fundamentalfeature of a rotation in the plane is that it is a free action on the boundary, i.e., there is no pointon the boundary S that remains fixed if we apply the rotation. This free action property ofrotations is crucial in the proof of k -Polygon Borsuk-Ulam Theorem and without this propertythe theorem does not hold. Unfortunately, in higher dimensions the rotations with respect to anyaxis no longer possess this property, as they always have a fixed point on the boundary S m − ofthe m -dimensional ball B m . Thus, in higher dimensions other operations acting freely on S m − emerge.For the case of k =
2, there is a very simple generalization of the rotation by 180 ◦ that is afree action in any number of dimensions, namely, the point reflection with respect to the origin , i.e., x (cid:55)→ − x . Hence, we have the following informal statement of the Borsuk-Ulam Theorem for ageneral number of dimensions. 7 nformal Theorem 5 (B orsuk -U lam T heorem ) . Let f : B m → R m be a continuous function that onthe boundary is equivariant to point reflection with respect to the origin. Then, there exists x (cid:63) ∈ B m suchthat f ( x (cid:63) ) = . Observe that the order of the point reflection operation is equal to k =
2, since if we apply thesame operation twice, we return to the same point. It is also trivial to see that the rotation by360 ◦ / k degrees has order k , since after k times of applying this operation we return to the samepoint and if we apply this operation less than k times, then we end up on a different point. Theseobservations together with the requirement for a free action suggest that in order to generalizethe k -Polygon Borsuk-Ulam theorem to higher dimensions we need a free action of order k on theboundary S m − of the m -dimensional ball B m . Unfortunately, finding such operations is not aseasy as in the case k =
2. In particular, for m = (cid:96) + S (cid:96) has a free action of order k only for k = α p that has order p , where p is a prime number, and defines a free action on the sphere S n ( p − ) − .These restrictions on α p are the reason why, as we mentioned in the beginning of the section, BSSonly applies to dimensions that are multiples of p − k = p where p is a prime number.Using the operation α p , we can informally state the BSS Theorem as follows: Informal Theorem 6 (BSS T heorem [Bárány, Shlosman, and Szücs, 1981]) . Let f : B n ( p − ) → R n ( p − ) be a continuous function that is equivariant with respect to α p on the boundary. Then, there exists x (cid:63) ∈ B n ( p − ) such that f ( x (cid:63) ) = . The original proof of the BSS Theorem [Bárány et al., 1981] goes through the definition of indicesof functions in algebraic topology. One of our main contributions is to provide a combinatorialproof of the BSS Theorem. As in the case of k -Polygon Borsuk-Ulam, we provide a combinatorialproof of the BSS Theorem via the corresponding version of Tucker’s Lemma, which we call BSSTucker’s Lemma and we informally state below. Informal Theorem 7 (BSS T ucker ’ s L emma ) . Let p be a prime and T be a triangulation of B n ( p − ) with an α p -symmetric boundary. Suppose that every vertex x ∈ T has a color λ ( x ) ∈ Z p × [ n ] such that λ is equivariant with respect to α p on the boundary. Then, there exists a ( p − ) -simplex in T that has allthe colors ( j ) , . . . ( p , j ) for some j ∈ [ n ] . Our main technical contribution in this section is to show the following statements. (cid:46)
The computational problem that is associated with BSS Tucker’s Lemma is polynomial timeequivalent to the computational problem associated with the BSS Theorem (Theorem 4.8). (cid:46)
The computational problem associated with p -Polygon Tucker’s Lemma is reducible to thecomputational problem associated with BSS Tucker’s Lemma (Theorem 4.10). (cid:46) The computational problem associated with BSS Tucker’s Lemma is reducible to the compu-tational problem associated with Z p -star Tucker’s Lemma (Proposition 5.4). Z p -star Tucker’s Lemma is an existence theorem that we define informally in the next sectionand is the basic building block for proving the membership of Necklace Splitting in PPA- p . Aswe will see in the next section, we prove that the computational problem associated with Z p -starTucker’s Lemma is inside PPA- p . This result combined with Informal Theorem 4, which showsthe PPA- p -completeness of the p -Polygon Borsuk-Ulam Theorem, and the equivalence of the p -Polygon Borsuk-Ulam Theorem and p -Polygon Tucker’s lemma, implies the main result of thissection. 8 nformal Theorem 8. The computational problem associated with the BSS Theorem is
PPA- p-complete.
We defer the formal definition of the computational problem associated with the BSS Theoremand the complete proofs, including the proof of Informal Theorem 8, about the BSS Theoremappear in Section 4. p thieves is in PPA- p Our main goal in the last part of the paper is to show that the computational problems associatedwith the p -Necklace Splitting Theorem and the Consensus-1/ p -Division Theorem are in PPA- p .Towards this goal we provide a full combinatorial proof for both of these problems that simplifiesthe only existing combinatorial proof by Meunier [2014].Our proof uses a different generalization of Tucker’s Lemma which we call Z p -star Tucker’sLemma. Both the statement of Z p -star Tucker Lemma and the proof are simple enough so thatthey do not invoke the involved definition of simplotopal complexes of Meunier [2014]. Thesesimplotopal complexes were used by Meunier in order to obtain a direct proof of the Necklace-Splitting theorem, without proving its continuous version. In contrast, we use standard simplicialcomplexes, since we are not only interested in the Necklace Splitting theorem, but also in itscontinuous generalization: the Consensus-1/ k -Division Theorem (informally defined below).Recall that in the k -Necklace Splitting problem, k thieves are aiming to split an open necklacecontaining n beads of t different colors, with exactly k · a i beads of color i , for some a i ∈ N , suchthat each thief receives exactly a i beads of color i . The k -Necklace Splitting Theorem states thefollowing. Informal Theorem 9 ( k -N ecklace S plitting T heorem [Alon, 1987]) . The k-Necklace Splittingproblem always has a solution with ( k − ) t cuts. The Consensus-1/ k -Division problem resembles the continuous version of the k -Necklace Splittingproblem. In this problem, each one of t agents has a probability measure µ i over the unit interval [
0, 1 ] . The goal is to cut the interval [
0, 1 ] into pieces and assign one of k possible colors to eachpiece such that every agent measures the total mass of each different color the same. Informal Theorem 10 (C onsensus -1/ k -D ivision T heorem [Alon, 1987, Simmons and Su, 2003]) . The Consensus- k-Division problem always has a solution with ( k − ) t cuts. Both the k -Necklace Splitting Theorem and the Consensus-1/ k -Division Theorem have significantapplications to combinatorics and social choice; see Section 1.2.As we have already mentioned, the proof that for prime p the computational problemsassociated with the p -Necklace Splitting Theorem and the Consensus-1/ p -Division Theorem areinside PPA- p , is based on a different generalization of Tucker’s Lemma for modulo- p arguments, Z p -star Tucker’s Lemma. The main difference of Z p -star Tucker’s Lemma with BSS Tucker’sLemma is the domain on which we define the triangulation. In the case of BSS Tucker’s Lemma,the triangulation is defined over a convex domain and hence is homeomorphic with the ball B m .On the other hand, the triangulation for Z p -star Tucker’s Lemma is defined over a star-convex setwhich is not homeomorphic to the ball B m anymore. This d -dimensional domain, which we denoteby R dp , is a slightly modified version of the domain used by Meunier [2014] in his combinatorialproof of the p -Necklace Splitting Theorem. It admits a natural free action θ p of order p . We caninformally state Z p -star Tucker’s Lemma similarly to BSS Tucker’s lemma as follows.9 nformal Theorem 11 ( Z p - star T ucker ’ s L emma ) . Let p be a prime and T be a triangulation ofR t ( p − ) p . Suppose that every vertex x ∈ T has a color λ ( x ) ∈ Z p × [ t ] such that λ is equivariant withrespect to θ p on the boundary. Then, there exists a ( p − ) -simplex of T that has all the colors ( j ) , . . . ( p , j ) for some j ∈ [ t ] . Our main technical contributions in this section are summarized in the following statements. (cid:46)
The computational problems that are associated with the p -Necklace Splitting Theorem andthe Consensus-1/ p -Division Theorem are polynomial time reducible to Z p -star Tucker’sLemma (Theorem 6.2). (cid:46) The computational problem associated with Z p -star Tucker’s Lemma is inside PPA- p (The-orem 5.3). This is again proved by a reduction to I mbalance - mod - p , but this time weconstruct a weighted directed graph. (cid:46) The computational problem associated with BSS Tucker’s Lemma is polynomial time re-ducible to the computational problem associated with Z p -star Tucker’s Lemma (Proposi-tion 5.4).The above statements combined with the results in the previous sections imply our main resultfor this part of the paper. Informal Theorem 12.
The computational problem associated with Z p -star Tucker’s Lemma is PPA- p-complete. The computational problems that are associated with the p-Necklace Splitting Theorem and theConsensus- p-Division Theorem are in PPA- p. As a corollary of this result, we also obtain that for general k ≥ k -Necklace Splitting andConsensus-1/ k -Division lie in PPA- k under Turing reductions . In particular, if k = p r is a primepower, then the corresponding problems lie in PPA- p . Furthermore, our reductions also provide anew combinatorial proof of the Necklace Splitting theorem, that is conceptually simpler and doesnot use any involved machinery.The formal definitions and the complete proofs, including the proof of Informal Theorem 12,about Z p -star Tucker’s Lemma, the p -Necklace Splitting Theorem and the Consensus-1/ p -DivisionTheorem appear in Section 5 and Section 6.Our results are summarized in Table 1, where we also highlight where they fit in the computationallandscape of the classes of interest. Relevant further related work is discussed in the next section. Computational Classes:
As mentioned earlier, among the classes of TFNP, PPAD has been themost successful in capturing the complexity of well-known problems. Besides the complexity ofcomputing a Nash equilibrium [Daskalakis et al., 2009, Chen et al., 2009b, Rubinstein, 2018], otherPPAD-complete problems are computing equilibria in markets [Chen et al., 2009a, 2013], versionsof envy-free cake cutting [Deng et al., 2012] and fixed-point theorems [Mehta, 2014, Goldberg andHollender, 2019].For PPA, the recent results by Filos-Ratsikas and Goldberg [2018, 2019] have solidified thestatus of the class as one that contains natural problems. In particular, they showed that 2-thiefNecklace Splitting is PPA-complete; the proof goes via its continuous version, the
Consensus-Halving problem of Simmons and Su [2003] Our PPA- p -membership result for the problem with The hardness result of Filos-Ratsikas and Goldberg [2018, 2019] for the Consensus-Halving problem was strength-ened recently by Filos-Ratsikas et al. [2020] to the case of simpler measures. PAD PPA PPA- p ( p ≥ ) TopologicalExistenceTheorem B rouwer [Papadimitriou, 1994][Chen and Deng, 2009] H airy -B all [Goldberg and Hollender, 2019] B orsuk -U lam [Papadimitriou, 1994][Aisenberg et al., 2020] p - polygon -B orsuk -U lam p -BSS [This Work] CombinatorialLemma S perner [Papadimitriou, 1994][Chen and Deng, 2009] T ucker [Papadimitriou, 1994][Aisenberg et al., 2020] p - polygon -T ucker p -BSS-T ucker Z p - star -T ucker [This Work] NotableProblems N ash [Daskalakis et al., 2009][Chen et al., 2009b] M arket -E quilibrium [Chen et al., 2009a, 2013] and many more... ecklace -S plitting C onsensus -H alving D iscrete -H am -S andwich [Filos-Ratsikas and Goldberg, 2019] S ymmetric -C hevalley - mod - p [Göös et al., 2020] p -N ecklace -S plitting C onsensus -1/ p -D ivision [Membership: This Work][Hardness: Open] Table 1: An overview of the computational landscape for the related TFNP classes. p thieves also uses the continuous variant, termed as Consensus- p-Division in [Simmons andSu, 2003]; we note that Alon [1987] used the same problem in his existence proof, referring to itas a generalized Hobby-Rice Theorem . As we explained earlier, Filos-Ratsikas and Goldberg [2019]conjectured that the problem with p thieves is complete for PPA- p .The classes PPA- p were introduced by Papadimitriou [1994] for any prime p , in the context ofclassifying a computational version of the Chevalley-Warning theorem [Chevalley, 1935, Warning,1935]. He proved that the corresponding problem C hevalley - mod - p lies in PPA- p . Recently,Göös et al. [2020] showed that an explicit version of the problem is complete for PPA- p , thereforeobtaining the first PPA- p -completeness result for a natural problem. The authors of [Göös et al.,2020], as well as Hollender [2019], independently also extended the definition of the classesPPA- k to any k ≥
2, and provided several characterizations in terms of their defined-for-primescounterparts. Hollender [2019] also investigated the connection with the classes PMOD- k , whichbear strong resemblance to PPA- k , and were defined seemingly independently of Papadimitriou’swork by Johnson [2011]. From the work of Hollender [2019], we primarily make use of the PPA- k -complete computational problem I mbalance - mod - k , a generalization of the PPAD-completeproblem I mbalance [Beame et al., 1998, Goldberg and Hollender, 2019]. Necklace Splitting:
The origins of the Necklace Splitting problem (Informal Theorem 9), and itscontinuous variant (Informal Theorem 10), can be traced back to work of Neyman [1946] andHobby and Rice [1965]. The problem was firstly phrased as a necklace splitting problem by Bhattand Leiserson [1982]. Later on, Goldberg and West [1985] and Alon and West [1986] provedthe first existence results for two thieves, and Alon [1987] extended the result to the case of anynumber k ≥ The BSS Theorem:
The BSS Theorem (Informal Theorem 6, Theorem 4.2), due to [Bárány et al.,1981], is perhaps the most well-known generalization of the Borsuk-Ulam theorem. Besides [Alon,1987], it has been used to prove existence of other interesting problems, including a generalizationof the Kneser-Lovász Theorem [Kneser, 1955, Lovász, 1978] due to Alon et al. [1986], a generalizedvan Kampen-Flores Theorem [Sarkaria, 1991] and the generalization to Tverberg’s Theorem,proven in [Bárány et al., 1981]. We believe that our PPA- p -completeness result paves the way forstudying the complexity of those problems as well. Let {
0, 1 } ∗ denote the set of all finite length bit-strings. For x ∈ {
0, 1 } ∗ , let | x | be its length. Acomputational search problem is given by a binary relation R ⊆ {
0, 1 } ∗ × {
0, 1 } ∗ . The associatedproblem is: given an instance x ∈ {
0, 1 } ∗ , find a y ∈ {
0, 1 } ∗ such that ( x , y ) ∈ R , or return thatno such y exists. The search problem R is in FNP ( Functions in NP ), if R is polynomial-timecomputable (i.e., ( x , y ) ∈ R can be decided in polynomial time in | x | + | y | ) and there exists somepolynomial p such that ( x , y ) ∈ R = ⇒ | y | ≤ p ( | x | ) . Thus, FNP is the search problem version ofNP.The class TFNP ( Total Functions in NP [Megiddo and Papadimitriou, 1991]) contains all FNPsearch problems R that are total : for every x ∈ {
0, 1 } ∗ there exists y ∈ {
0, 1 } ∗ such that ( x , y ) ∈ R .Note that the totality of problems in TFNP does not rely on any “promise”. Instead, there is a syntactic guarantee of totality: for any instance in {
0, 1 } ∗ , there is always at least one solution.Let R and S be total search problems in TFNP. We say that R (many-one) reduces to S , denoted R ≤ S , if there exist polynomial-time computable functions f , g such that ( f ( x ) , y ) ∈ S = ⇒ ( x , g ( x , y )) ∈ R .Note that if S is polynomial-time solvable, then so is R . We say that two problems R and S are(polynomial-time) equivalent, if R ≤ S and S ≤ R .Sometimes a more general notion of reduction is used. A Turing reduction from R to S isa polynomial-time oracle Turing machine that solves problem R with the help of queries to anoracle for S . Note that a Turing reduction that only makes a single oracle query immediatelyyields a many-one reduction. 12 .2 The Classes PPA- k For k ≥
2, PPA- k is a subclass of TFNP that aims to capture the complexity of TFNP problemswhose totality is proved by using an argument modulo k . The classes PPA- p (for prime p ) wereintroduced by Papadimitriou [1994]. The case p = parity arguments , and in thatcase the class PPA-2 is simply called PPA. Recently, the definition of PPA- k was extended to any k ≥ k is defined as the class of TFNPproblems that reduce to the following problem. Definition 1 (B ipartite - mod - k [Papadimitriou, 1994]) . Let k ≥
2. The problem B ipartite - mod - k is defined as: given a Boolean circuit C : {
0, 1 } × {
0, 1 } n → ( {
0, 1 } × {
0, 1 } n ) k that computes abipartite graph on the vertex set ( { } × {
0, 1 } n , { } × {
0, 1 } n ) with | C ( n ) | ∈ {
1, . . . , k − } , find• x (cid:54) = n such that | C ( x ) | / ∈ { k } • or x , y such that y ∈ C ( x ) but x / ∈ C ( y ) .The circuit C computes a bipartite graph as follows. The output of circuit C on some input x is a list of k bit-strings, each of length n +
1. If x ∈ { } × {
0, 1 } n , then we let C ( x ) denotethe set of distinct bit-strings that appear in that list and that lie in { } × {
0, 1 } n . Similarly, if x ∈ { } × {
0, 1 } n , then C ( x ) denotes the set of distinct bit-strings that appear in that list and thatlie in { } × {
0, 1 } n . For any x , y ∈ {
0, 1 } × {
0, 1 } n , there exists an edge between x and y if andonly if y ∈ C ( x ) and x ∈ C ( y ) . It is easy to see that this indeed defines a bipartite graph. Definition 2.
Let k ≥ ≤ (cid:96) ≤ k −
1. The problem B ipartite - mod - k [ (cid:96) ] is defined asB ipartite - mod - k (Definition 1) but with the additional condition | C ( n ) | = (cid:96) .Note that this condition can be enforced syntactically and so this problem also lies in TFNP (see[Papadimitriou, 1994] for a definition of “syntactic”). Definition 3 (PPA- k [ (cid:96) ] ) . Let k ≥ ≤ (cid:96) ≤ k −
1. The class PPA- k [ (cid:96) ] is defined as the set ofall TFNP problems that many-one reduce to B ipartite - mod - k [ (cid:96) ] .The following result relates these special subclasses to the main ones. Proposition 2.1 ([Hollender, 2019]) . PPA- k [ ] = ∩ p ∈ PF ( k ) PPA- p, where PF ( k ) denotes the set ofprime factors of k. In this paper, we will also use the following definition of PPA- k , which was shown to be equivalentto the standard one [Hollender, 2019]. The class PPA- k is the set of all TFNP problems that reduceto the following problem. Definition 4.
Let k ≥
2. The problem I mbalance - mod - k is defined as: given Boolean circuits S , P : {
0, 1 } n → ( {
0, 1 } n ) k with | S ( n ) | − | P ( n ) | (cid:54) = k , find• x (cid:54) = n such that | S ( x ) | − | P ( x ) | (cid:54) = k • or x , y such that y ∈ S ( x ) but x / ∈ P ( y ) , or y ∈ P ( x ) but x / ∈ S ( y ) .For 1 ≤ (cid:96) ≤ k −
1, I mbalance - mod - k [ (cid:96) ] is defined with the additional condition | S ( ) | − | P ( ) | = (cid:96) . 13e obtain a seemingly more general version of this problem by allowing the edges to have integerweights. In that case the imbalance of a vertex is measured as the difference of the weights of allincoming edges and the weights of all outgoing edges. It is easy to see that this problem is in factequivalent to I mbalance - mod - k . First, all the weights can be assumed to be in {
0, 1, . . . , k − } ,since we can reduce them modulo k . Next, we can split an edge with weight (cid:96) into (cid:96) copies of theedge. Finally, to ensure that we don’t have multi-edges, we add a new vertex in the middle ofevery edge. Note that the new vertex will be balanced by construction, and will thus not introduceany new solutions.It is easy to see that in all the aforementioned problems a solution is always guaranteed toexist. The search problems are not trivial though, because the graph can have exponential sizewith respect to its description. Indeed, the graph is given by Boolean circuits that compute thesuccessors and predecessors of every vertex.We make use of the following known properties of these classes: Proposition 2.2 (Göös et al. [2020], Hollender [2019]) . It holds that: • for any prime p and any r ≥ , PPA- p r = PPA- p • for any k , (cid:96) ≥ , PPA- k ⊆ PPA- k (cid:96) Topological Definitions and Details:
Our results in the next sections will require several def-initions from topology, as well as the corresponding notation. The more experienced readermight already be familiar with some of these concepts, but all the relevant details are included inAppendix A. There, we also define our value and index functions, which allow us to enumerateover simplices, and access simplices that contain a point x in the domain respectively. k -Polygon Borsuk-Ulam: a PPA- k [ ] -complete Problem in 2D-space In this section we present a generalization of the Borsuk-Ulam Theorem in two dimensional space.Surprisingly this theorem is not captured by the BSS Theorem and hence has its own topologicalinterest. Our proof of this theorem is combinatorial and is based on a generalization of Tucker’sLemma which we call
Polygon Tucker’s Lemma , hence it is very different from the proof of the BSSTheorem. Our main result is that k -Polygon Borsuk-Ulam is complete for PPA- k [ ] . This givesthe first topological characterization of the classes PPA- k [ ] but also reveals in a very intuitiveway the relation between the different PPA- k [ ] classes and PPAD. Recall that when k = p r is aprime power, PPA- k [ ] = PPA- k = PPA- p .We start this section with a unified description of Brouwer’s Fixed Point Theorem, the Borsuk-Ulam Theorem and our generalization: the k -Polygon Borsuk-Ulam Theorem. For this we willneed the following definition. Definition 5 (R otation O perator ) . We define the k -th rotation operator θ k : R → R as follows: θ k ( x ) = R k x , where R k is the following two dimensional rotation matrix R k = (cid:20) cos (cid:0) − π k (cid:1) − sin (cid:0) − π k (cid:1) sin (cid:0) − π k (cid:1) cos (cid:0) − π k (cid:1) (cid:21) .In other words, θ k corresponds to a clockwise rotation by an angle of 2 π / k .We continue with a statement of Brouwer’s Fixed Point Theorem that is different from thestandard statement, but is well known to be equivalent to that (e.g., see [Matoušek, 2008]).14 heorem 3.1 (B rouwer ’ s F ixed P oint T heorem ) . Let f : B → R be a continuous function such thatf ( x ) = x for all x ∈ ∂ B . Then there exists x (cid:63) ∈ B such that f ( x (cid:63) ) = . Next, using the same language, we give a statement of the Borsuk-Ulam Theorem.
Theorem 3.2 (B orsuk -U lam T heorem ) . Let f : B → R be a continuous function such thatf ( θ ( x )) = θ ( f ( x )) for all x ∈ ∂ B . Then there exists x (cid:63) ∈ B such that f ( x (cid:63) ) = . It is clear from the above expression that the Borsuk-Ulam Theorem is a generalization ofBrouwer’s Fixed Point Theorem. This observation is in line with the fact that PPAD ⊆ PPA,since Brouwer is complete for PPAD and Borsuk-Ulam is complete for PPA. We now present ourextension, that we call k -Polygon Borsuk-Ulam Theorem. Theorem 3.3 ( k -P olygon B orsuk -U lam T heorem ) . Let f : B → R be a continuous function suchthat f ( θ k ( x )) = θ k ( f ( x )) for all x ∈ ∂ B . Then there exists x (cid:63) ∈ B such that f ( x (cid:63) ) = . As we will see the k -Polygon Borsuk-Ulam Theorem is also a generalization of Brouwer’sFixed Point Theorem and it is complete for PPA- k [ ] which is also in line with the fact thatPPAD ⊆ PPA- k [ ] . Another interesting fact about the k -Polygon Borsuk-Ulam Theorem is that itdoes not directly follow from the traditional generalization of the Borsuk-Ulam Theorem, namelythe BSS Theorem, as we will see in the next section. k -Polygon Tucker’s Lemma and k -Polygon Borsuk-Ulam in PPA- k [ ] In this section, we define k -Polygon Tucker’s Lemma and we prove that it is equivalent to the k -Polygon Borsuk-Ulam Theorem. Additionally, we provide a combinatorial proof of k -PolygonTucker’s Lemma using a modulo- k argument. This combinatorial proof puts both k -PolygonTucker and k -Polygon Borsuk-Ulam in the class PPA- k [ ] .Before showing the equivalence of the two statements, we provide some necessary notationand definitions. Definition 6 ( k -P olygon & N ice T riangulation ) . For k ≥
3, let W k be the regular k -polygon, i.e.,regular k -gon, centered at ∈ R with radius 1. Let u , . . . , u k denote the vertices of W k , orderedsuch that θ k ( u i ) = u i + ( mod k ) for all i ∈ [ k ] . We define T (cid:63) to be the triangulation of W k thatincludes the simplices σ i = conv ( { , u i , u i + ( mod k ) } ) for i ∈ [ k ] . We call a triangulation T nice if itsatisfies the following two properties:• it is a refinement of T (cid:63) , and• it is symmetric with respect to θ k on the boundary. This means that for every edge ψ ∈ T such that ψ ⊆ ∂ W k it holds that θ k ( ψ ) ∈ T . Theorem 3.4 ( k -P olygon T ucker ’ s L emma ) . For k ≥ , let W k be a k-regular polygon. Fix some nicetriangulation T of W k . Suppose that every vertex x ∈ T has a label λ ( x ) ∈ Z k such that for any y ∈ ∂ Twe have λ ( θ k ( y )) = λ ( y ) + ( mod k ) . Then at least one of the following exists: (1) a simplex σ ∈ Twith vertices v , v , v such that all the labels λ ( v ) , λ ( v ) , λ ( v ) are different, or (2) an edge ψ ∈ Twith vertices v , v such that λ ( v ) − λ ( v ) ( mod k ) / ∈ {
0, 1, − } .Remark . The above theorem can be proved if we invoke Dold’s Theorem from algebraic topologyDold [1983]. However this proof is not a constructive proof and hence it cannot be used for our15urposes and for this reason we reprove this theorem using a constructive combinatorial proof.Of course, this is only a special case of the general Dold’s Theorem and it is an interesting openproblem what is the relation of Dold’s Theorem with the subclasses of TFNP as we state in theConclusions section. k -Polygon Tucker and k -Polygon Borsuk-Ulam We start by showing that k -Polygon Tucker’s Lemma is implied by k -Polygon Borsuk-UlamTheorem and then we also show the converse, in Lemma 3.5 and Lemma 3.6. Lemma 3.5. k-Polygon Borsuk-Ulam (Theorem 3.3) implies k-Polygon Tucker’s Lemma (Theorem 3.4).Proof.
We interpret each label i ∈ Z k as the vector u i , which is the i -th vertex of the polygon W k .Let h : W k → W k be the piecewise linear extension of the function that has value u λ ( x ) on anyvertex x ∈ T . Finally we define g : B → R as the composition of h and a homeomorphismbetween W k and B that maps to and is θ k -equivariant. Notice that by the definition of h and g and the equivariance assumption on λ , it holds that g ( θ k ( x )) = θ k ( g ( x )) for x ∈ ∂ B . Then, itfollows from Theorem 3.3 that there exists an y (cid:63) ∈ B such that g ( y (cid:63) ) = x (cid:63) ∈ W k such that h ( x (cid:63) ) = σ (cid:63) be a full-dimensional simplex of T that contains x (cid:63) . If the vertices of σ (cid:63) have onlytwo consecutive labels, say 1 and 2 (corresponding to vectors u and u ), then it is impossibleto have h ( x (cid:63) ) = u and u and these vectors arelinearly independent. Hence, it has to be that the vertices of σ (cid:63) have either two non-consecutivelabels or three different labels and k -Polygon Tucker’s Lemma follows. Lemma 3.6. k-Polygon Tucker’s Lemma (Theorem 3.4) implies k-Polygon Borsuk-Ulam (Theorem 3.3).Proof.
Let h : W k → R be the function obtained from f by using a homeomorphism between W k and B that fixes and is θ k -equivariant. Using standard arguments, that are used in the proof ofboth Brouwer’s Fixed Point Theorem via Sperner’s Lemma and the proof of Borsuk-Ulam viaTucker’s Lemma, it is enough if for every ε > x such that (cid:107) h ( x ) (cid:107) ≤ ε , for detailswe refer to [Matoušek, 2008]. Since h is a continuous function in a compact set, it is also uniformlycontinuous. Thus for every ε > δ > x , x (cid:48) ∈ W k , if (cid:107) x − x (cid:48) (cid:107) < δ ,then (cid:107) h ( x ) − h ( x (cid:48) ) (cid:107) < ε / k . Assume that T is a nice triangulation of W k such that any simplex σ ∈ T has diameter at most δ .We define the labeling λ ( x ) = i to be equal to the index of the vertex u i of W k that is closest to h ( x ) . We break ties between i and i +
1, by picking i . Note that the only other kind of tie that canoccur is if h ( x ) =
0. In that case all labels are tied. If x =
0, we pick one arbitrarily. Otherwise, i.e.,if x (cid:54) =
0, we apply the same rule as for h ( x ) above, but for x instead. It is easy to check that thistie-breaking is Z k -equivariant. For any x ∈ ∂ W k because of the equivariance assumption on g wehave that λ ( θ k ( x )) = λ ( x ) + ( mod k ) and hence the assumptions of k -Polygon Tucker’s Lemmaare satisfied. Now we distinguish two cases. k = In this case, 3-Polygon Tucker’s Lemma implies that there exists a simplex σ ∈ T whosevertices v , v , v contain all the labels 1, 2, and 3 respectively. This implies the following set ofinequalities by the definition of the labeling λ (cid:107) h ( v ) − u (cid:107) ≤ min ( (cid:107) h ( v ) − u (cid:107) , (cid:107) h ( v ) − u (cid:107) ) (cid:107) h ( v ) − u (cid:107) ≤ min ( (cid:107) h ( v ) − u (cid:107) , (cid:107) h ( v ) − u (cid:107) ) h ( v ) − u (cid:107) ≤ min ( (cid:107) h ( v ) − u (cid:107) , (cid:107) h ( v ) − u (cid:107) ) .Hence, it is easy to see that the maximum angle between any of h ( v ) , h ( v ) and h ( v ) is at least2 π /3. Now, assume for the sake of contradiction that for all v , v , v it holds that (cid:107) h ( v ) (cid:107) ≥ ε , (cid:107) h ( v ) (cid:107) ≥ ε , (cid:107) h ( v ) (cid:107) ≥ ε .This together with the fact that the maximum angle is at least 2 π /3 implies that the maximumdistance is at least 2 sin ( π /6 ) ε . But this implies thatmax ( (cid:107) h ( v ) − h ( v ) (cid:107) , (cid:107) h ( v ) − h ( v ) (cid:107) , (cid:107) h ( v ) − h ( v ) (cid:107) ) ≥ √ · ε > ε T , where the vertices of the same simplexare at most δ far from each other and hence their images are at most ε / k = ε /3 far from eachother. This implies that our assumption was wrong and hence for at least one i ∈ [ ] it holds that (cid:107) h ( v i ) (cid:107) ≤ ε and the result for this case follows. k > In this case, k -Polygon Tucker’s Lemma implies that there exists an edge ψ ∈ T whosevertices v and v have labels that differ by more that one, without loss of generality assume thatthese labels are 1 and 3 respectively. By the definition of the labeling λ this implies that (cid:107) h ( v ) − u (cid:107) ≤ min i ( (cid:107) h ( v ) − u i (cid:107) ) , (cid:107) h ( v ) − u (cid:107) ≤ min i ( (cid:107) h ( v ) − u i (cid:107) ) Hence, it is easy to see that the angle between h ( v ) and h ( v ) is at least 2 π / k . Now, for sake ofcontradiction we assume that (cid:107) h ( v ) (cid:107) ≥ ε , (cid:107) h ( v ) (cid:107) ≥ ε .This together with the fact that the angle is at least 2 π / k implies that the distance (cid:107) h ( v ) − h ( v ) (cid:107) is at least 2 sin ( π / k ) ε > ε / k which contradicts the definition of T .Therefore, in both cases for every ε > v ∈ W k such that (cid:107) h ( v ) (cid:107) ≤ ε . This, bystandard arguments and the compactness of W k , implies that there exists a point x (cid:63) ∈ W k suchthat h ( x (cid:63) ) = and hence the result follows. k -Polygon Tucker’s Lemma In this section, we prove k -Polygon Tucker’s Lemma. A corollary of our proof combined withLemma 3.6 is that the computational problems associated with k -Polygon Tucker’s Lemma and k -Polygon Borsuk-Ulam both belong to PPA- k [ ] . The proof technique that we introduce here isa generalization of the combinatorial proof of Tucker’s lemma given by Freund and Todd [1981].We use a modulo- k argument to prove this theorem. We define a directed graph where thevertices correspond to the simplices of T , and we also identify the symmetric edges of T on theboundary of W k as the same vertex. Then, in this graph we describe a rule for defining edges suchthat there exist three types of vertices:1. the vertex that corresponds to the 0-dimensional simplex { } which will have degree 1,2. vertices that are balanced ( mod k ) , i.e., ( out - degree ) − ( in - degree ) = ( mod k ) ,3. vertices that are different from { } and are not balanced ( mod k ) .17igure 1: Example that shows happy and non-happy simplices in the cone conv ( { u , , u } ) .Due to a simple modulo- k argument and because { } is not balanced we can conclude thatthe constructed graph contains a vertex of type 3. Finally, we prove that all vertices of type 3correspond to either a trichromatic triangle or a bichromatic edge with distinct non-consecutivelabels and hence k -Polygon Tucker’s Lemma follows. Our proof also gives us a reduction of thecomputational problem associated with k -Polygon Tucker’s Lemma to the problem I mbalance - mod - k [ ] .For any simplex σ ∈ T we define S ( σ ) and λ ( σ ) : (cid:46) S ( σ ) ⊆ [ k ] is the minimal subset of [ k ] such that σ lies in the cone defined by { u i : i ∈ S ( σ ) } , (cid:46) λ ( σ ) = { λ ( x ) : x is a vertex of σ } ,and we let S ( { } ) = ∅ . Remark . Observe that because T is a refinement of T ∗ , we have that every simplex σ ∈ T iscontained in a cone defined by two consecutive vectors u i , u i + . Hence, for σ (cid:54) = { } , S ( σ ) containseither a single number i ∈ [ k ] or two consecutive numbers i , i + Definition 7 (H appy S implices ) . A simplex σ ∈ T is called happy if and only if S ( σ ) ⊆ λ ( σ ) . Seealso Figure 1 for an example that explains the definition.We will define a graph G with vertex set V ( G ) = T . In G , we will only add directed edges tothe following vertices, which we call relevant :(a) vertices that correspond to a simplex σ ∈ T such that σ (cid:54)∈ ∂ T and σ is happy,(b) vertices that correspond to a simplex ψ ∈ ∂ T such that ψ is happy and: – ψ = { u } , if ψ is 0-dimensional – ψ ⊆ conv ( { u , u } ) , if ψ is 1-dimensional.The reason for this distinction between simplices on the boundary and simplices not on theboundary is that we want to identify the symmetric simplices on the boundary as a super vertex inorder to correctly use a modulo- k argument, as we described in the sketch of the proof.18 emark . The rest of the vertices in V ( G ) that do not correspond to type (a) or (b) simplices canbe thought of as having ( out - degree ) = ( in - degree ) = ( v , v (cid:48) ) to the graph G only if the simplices σ and σ (cid:48) that correspond to v and v (cid:48) are both relevant and one of the following rules applies:1. case σ (cid:54)∈ ∂ T , σ (cid:48) (cid:54)∈ ∂ T : We add the edge if σ (cid:48) ⊆ σ and the labels of σ (cid:48) suffice to make σ happy, i.e., S ( σ ) ⊆ λ ( σ (cid:48) ) ,2. case σ (cid:54)∈ ∂ T , σ (cid:48) ∈ ∂ T : Observe that since v (cid:48) is relevant and σ (cid:48) ∈ ∂ T , v (cid:48) is of type (b).So, instead of checking whether σ (cid:48) ⊆ σ , we check whether there exists t ∈ [ k ] such that τ : = θ ( t ) k ( σ (cid:48) ) ⊆ σ and τ suffices to make σ happy, i.e., S ( σ ) ⊆ λ ( τ ) . Directing the edges.
The edge between σ and σ (cid:48) is directed in the following natural way:• if σ is 1-dimensional, then σ and σ (cid:48) lie in conv ( { u i } ) for some i , and the edge is directed“away from 0”. Formally, if σ = { z , z } and σ (cid:48) = { z } are connected by an edge, then write z = α i u i and z = β i u i . If α i − β i >
0, then the edge is incoming into σ . If α i − β i <
0, thenthe edge is outgoing out of σ .• if σ is 2-dimensional, then σ and σ (cid:48) lie in conv ( { u i , u j } ) for some i , j with i − j = ± ( mod k ) . If j = i +
1, then the edge is directed such that “we keep label i to our right, andlabel j = i + j = i −
1, thenthe edge is directed such that “we keep label j = i − i to our left,when we move in the direction of the edge”. Formally, if σ = { z , z , z } and σ (cid:48) = { z , z } are connected by an edge, where λ ( z ) = i and λ ( z ) = j , then write z = α i u i + α j u j , z = β i u i + β j u j and z = γ i u i + γ j u j . Construct the matrix M = (cid:20) α i − β i α i − γ i α j − β j α j − γ j (cid:21) If det M >
0, then the edge is incoming into σ . If det M <
0, then the edge is outgoing outof σ . Notice that det M (cid:54) =
0, because σ is a simplex. Furthermore, note that the determinantof the matrix does not change if we switch both i and j , and z and z (i.e., β and γ ). Thus,the direction is well-defined.If σ (cid:48) corresponds to a vertex of type (b), then we apply the rule above to σ and τ instead.Based on the description of the edges in the graph G we can prove the following Lemmaswhich complete the proof of k -Polygon Tucker’s Lemma. Lemma 3.7.
The vertex that corresponds to the simplex { } ∈ T has out-degree and in-degree .Proof. Since { } is a 0-dimensional simplex, it does not have any sub-simplices and hence itcan only be connected to a 1-dimensional simplex, i.e., an edge. An edge ψ ∈ T that is notcontained in a linear segment of the form conv ( { , u i } ) cannot be a neighbor of { } , because itrequires two labels to become happy and { } has only one single label. Hence, { } can onlybe connected to a 1-dimensional simplex { , z i } contained in conv ( { , u i } ) for some i . Observealso that S ( { , z i } ) = { i } , which implies that S ( { , z i } ) = { λ ( ) } . Therefore, there is exactly one1-dimensional simplex that is connected to { } , namely { , z i } , where i = λ ( ) .19igure 2: An example of the graph G constructed in the proof of k -Polygon Tucker’s Lemma,focused on the cone conv ( { u , , u } ) .Furthermore, since { } and its neighbor { , z i } lie in conv ( { , u i } ) , the edge is directed awayfrom { } . Formally, if we write z i = α i u i and = β i u i , it will hold that α i − β i = α i >
0. Thismeans that the edge is incoming into { , z i } and the lemma follows. Lemma 3.8.
Any vertex v in G that is imbalanced modulo-k, i.e., ( out - degree ( v )) (cid:54) = ( in - degree ( v ))( mod k ) and does not correspond to the simplex { } , corresponds to a simplex σ that is either trichromaticor λ ( σ ) (cid:54)⊆ { i , i + } for all i ∈ [ k ] .Proof. For this proof, we consider all three cases for the dimension of the simplex σ (cid:63) that corre-sponds to an imbalanced node of G separately. Dimension of σ (cid:63) is 0. In this case, σ (cid:63) = { z (cid:63) } . It is easy to see that σ (cid:63) cannot make happy any2-dimensional simplex since a 2-dimensional simplex σ has | S ( σ ) | =
2. So, σ (cid:63) can only be aneighbor of a 1-dimensional simplex ψ . Additionally, σ (cid:63) cannot make happy any 1-dimensional ψ such that | S ( ψ ) | =
2. Thus, a neighbor of σ (cid:63) should be contained in the segment conv ( { , u i } ) where i is such that λ ( σ (cid:63) ) = i . If z (cid:63) (cid:54) = u i , then σ (cid:63) is connected to both of its two neighboring1-dimensional simplices in the segment conv ( { , u i } ) . Then, since the edges are directed awayfrom , σ (cid:63) has one incoming and one outgoing edge. Formally, let { z , z ∗ } and { z ∗ , z (cid:48) } be thetwo neighboring 1-dimensional simplices, and write z = α i u i , z (cid:48) = α (cid:48) i u i and z ∗ = β i u i . It is easyto see that α i − β i and α (cid:48) i − β i always have opposite signs, since z and z (cid:48) lie on opposite sides of z (cid:63) on conv ( { , u i } ) .If z (cid:63) = u i , then from the definition of relevant vertices of G we have that z (cid:63) = u and { u } is happy, i.e., λ ( u ) =
1. Thus, by the boundary conditions, for every t ∈ [ k ] , it holds that λ ( θ ( t ) k ( u )) = t +
1. In other words, λ ( u i ) = i for all i ∈ [ k ] . As a result, it follows that for each i ∈ [ k ] , { u } has an edge with the simplex { u i , z i } which lies in conv ( { , u i } ) . This holds because θ ( i ) k ( { u } ) suffices to make { u i , z i } happy. Clearly, θ ( i ) k ( { u } ) cannot make any other simplexhappy, by the same arguments as above. Finally, note that all the edges are incoming into { u } ,because edges are directed away from on any conv ( { , u i } ) . Formally, if we write z i = α i u i and u i = β i u i , then we always have α i − β i = α i − <
0, so the edge is outgoing out of { u i , z i } . Thus,20n this case, z (cid:63) has k neighbors and all of them with the same direction. Therefore, z (cid:63) is alwaysbalanced modulo- k . In conclusion, an imbalanced vertex of G different from { } cannot havedimension 0. Dimension of σ (cid:63) is 1. First, assume that σ (cid:63) = { z (cid:63) , z (cid:63) } belongs to one of the line segments conv ( { , u i } ) . If additionally λ ( z (cid:63) ) = λ ( z (cid:63) ) , then σ (cid:63) is happy if λ ( z (cid:63) ) = λ ( z (cid:63) ) = i . Therefore, | λ ( σ (cid:63) ) | = σ (cid:63) cannot be a neighbor of a 2-dimensional σ since | S ( σ ) | = σ (cid:63) cannot make σ happy. So, σ (cid:63) has exactly the two neighbors { z (cid:63) } and { z (cid:63) } since both of themmake σ (cid:63) happy. Furthermore, the vertex is balanced, because one of the edges is incoming andthe other one is outgoing, since edges are directed away from on conv ( { , u i } ) . Formally, if wewrite z (cid:63) = α i u i and z (cid:63) = α (cid:48) i u i , then α i − α (cid:48) i and α (cid:48) i − α i always have opposite signs.The next case we consider is when σ (cid:63) ⊆ conv ( { , u i } ) with i = λ ( z (cid:63) ) (cid:54) = λ ( z (cid:63) ) = j . If i − j (cid:54) = ±
1, then σ (cid:63) yields a solution to k -Polygon Tucker, and so is allowed to be imbalanced. Onthe other hand, if j = i ± ( mod k ) , σ (cid:63) has exactly two neighbors, the simplex { z (cid:63) } , which makes σ (cid:63) happy, and the unique 2-dimensional simplex σ = { z (cid:63) , z (cid:63) , z } that contains σ (cid:63) as a face and iscontained in the cone conv ( { u i , , u j } ) . Also, one of the edges is incoming and other one outgoingand hence σ (cid:63) cannot be imbalanced. Formally, write z (cid:63) = β i u i , z (cid:63) = β (cid:48) i u i and z = α i u i + α j u j .Then, it holds that the edge goes from { z (cid:63) } to { z (cid:63) , z (cid:63) } if β (cid:48) i − β i >
0, and otherwise in the otherdirection. Furthermore, the edge goes from σ = { z (cid:63) , z (cid:63) } to σ = { z (cid:63) , z (cid:63) , z } , if det M >
0, wherewe can compute that det M = ( α i − β i ) α j − ( α i − β (cid:48) i ) α j = α j ( β (cid:48) i − β i ) . Since α j >
0, it follows thatboth expressions have the same sign, and thus σ (cid:63) is balanced.The final case for 1-dimensional σ (cid:63) is when σ (cid:63) is not contained in any line segment of theform conv ( { , u i } ) . Let σ (cid:63) be contained in the cone conv ( { u i , , u i + } ) . If σ (cid:63) is not relevant, thenit cannot be imbalanced. To be relevant, and hence happy, it must hold that λ ( z (cid:63) ) = i and λ ( z (cid:63) ) = i +
1. If σ (cid:63) is not in the boundary ∂ T , then σ (cid:63) is the face of exactly two 2-dimensionalsimplices σ , σ (cid:48) that are both contained in conv ( { u i , , u i + } ) . Therefore, σ (cid:63) makes happy both ofthem and no other simplex, and consequently it has an edge with both of them and no other edge.Furthermore, one of the edges is incoming and other one is outgoing (from the perspective of σ (cid:63) ), so it cannot be imbalanced. Intuitively, if we let σ = { z (cid:63) , z (cid:63) , z } and σ (cid:48) = { z (cid:63) , z (cid:63) , z (cid:48) } , then z and z (cid:48) lie on opposite sides of the line defined by { z (cid:63) , z (cid:63) } . Thus, the basis of R defined by { z − z (cid:63) , z − z (cid:63) } has opposite orientation compared to the basis { z (cid:48) − z (cid:63) , z (cid:48) − z (cid:63) } . As a result,det M will have a different sign for the two edges. For a formal proof of this, we refer to the proofof Theorem 5.3, where we prove a more general version of this fact.If σ (cid:63) ∈ ∂ T , then σ (cid:63) is relevant only if σ (cid:63) ⊆ conv ( { u , u } ) . In this case, σ (cid:63) has exactly k neighbors each of which is a 2-dimensional simplex that has as a face one of the k symmetriccopies of σ (cid:63) . Namely, the neighbors of σ (cid:63) are σ , . . . , σ k , where θ ( i ) k ( σ (cid:63) ) makes σ i happy for all i ∈ [ k ] . To see that all k edges are incoming or all are outgoing, notice that if we have 1 to our rightand 2 to our left when we reach the boundary, then it will hold that we have i on our right and i + conv ( { , u i , u i + } ) .More formally, the sign of the determinant of the matrix M ( σ i , θ ( i ) k ( σ (cid:63) )) constructed to determinethe direction of the edge with σ i , will be the same for all i ∈ [ k ] . For a full formal proof, we againrefer to the proof of Theorem 5.3. Dimension of σ (cid:63) is 2. Let σ (cid:63) = conv ( { z (cid:63) , z (cid:63) , z (cid:63) } ) . Assume that σ (cid:63) is contained in the simplex conv ( { u i , , u j } ) , with j = i ± ( mod k ) and without loss of generality λ ( z (cid:63) ) = i , λ ( z (cid:63) ) = j . If λ ( z (cid:63) ) (cid:54)∈ { i , j } then σ (cid:63) is a trichromatic triangle, and thus it can be imbalanced. Assume that21 ( z (cid:63) ) ∈ { i , j } and without loss of generality λ ( z (cid:63) ) = i . In this case, σ (cid:63) has exactly two neighbors:the 1-dimensional simplices ψ = conv ( { z (cid:63) , z (cid:63) } ) and ψ = conv ( { z (cid:63) , z (cid:63) } ) . Once again, one edge isincoming and the other one is outgoing, and thus σ (cid:63) is balanced. Intuitively, this follows from thefact that if we move with i to our left and j to our right, then we can “enter” the simplex from oneside, and “exit” from the other one. More formally, if we write z (cid:63) = α i u i + α j u j , z (cid:63) = β i u i + β j u j and z (cid:63) = α (cid:48) i u i + α (cid:48) j u j , then it holds that:det (cid:34) α i − β i α i − α (cid:48) i α j − β j α j − α (cid:48) j (cid:35) = − det (cid:34) α (cid:48) i − β i α (cid:48) i − α i α (cid:48) j − β j α (cid:48) j − α j (cid:35) by using standard rules about the determinant.Hence, the only imbalanced vertices different from { } correspond to either trichromaticsimplices or simplices such that λ ( σ ) (cid:54)⊆ { i , i + } for all i ∈ [ k ] .Finally, from the definition of the graph G and Lemma 3.7, we have that there has to be avertex in G that is imbalanced ( mod k ) and different from { } . By Lemma 3.8, any such vertexproves the validity of k -Polygon Tucker’s Lemma. k [ ] In this section, we define the computational problems associated with k -Polygon Tucker’s Lemmaand the k -Polygon Borsuk-Ulam Theorem, which we call k - polygon -T ucker and k - polygon -B orsuk -U lam respectively. Following the ideas presented in Section 3.1.2, we show that k - polygon -T ucker is in PPA- k [ ] . The membership of k - polygon -T ucker in PPA- k [ ] combinedwith the results of Section 3.1.1 implies that k - polygon -B orsuk -U lam is also in PPA- k [ ] .To define the computational problem associated with k -Polygon Tucker’s Lemma we needsuccinct access to the labels λ ( x ) for the vertices x in a nice triangulation T of W k . This succinctaccess resembles the one in the definition of the computational version of the original Borsuk-UlamTheorem [Aisenberg et al., 2020, Papadimitriou, 1994] and the computational version of Brouwer’sFixed Point Theorem [Papadimitriou, 1994]. To define this succinct access, we fix for any m ∈ N atriangulation T ( m ) with diameter (i.e., max distance between two vertices of a simplex) at most1/2 m , where we can refer to a simplex in T ( m ) using O ( m + k ) bits. For our case this triangulationwill be the following. Definition 8 (E dge P arallel T riangulation ) . For every m ∈ N , we define (cid:98) T ( m ) to be thefollowing nice triangulation of W k . Starting from T (cid:63) (see Definition 6) we define a simplicial com-plex (cid:98) T i ( m ) of every simplex σ (cid:63) i = conv ( { u i , , u i + ( mod k ) } ) and then set (cid:98) T ( m ) = ∪ i ∈ [ k ] (cid:98) T i ( m ) .To define (cid:98) T i ( m ) , we divide the edges ψ i = conv ( { u i , } ) , ψ i = conv ( { u i , u i + ( mod k ) } ) and ψ i = conv ( { , u i + ( mod k ) } ) of σ (cid:63) i equally into 2 m + intervals. Then, from any endpoint ofthe subintervals of the edge ψ i we consider the lines that are parallel to either ψ i or ψ i and fromany endpoint of the subintervals in ψ i and ψ i we consider the line that is parallel to ψ i , as shownin Figure 2. We define (cid:98) T i ( m ) to be the set of simplices that are created by these lines and lie inside σ (cid:63) i . Namely, the intersection points and endpoints of subintervals are the 0-dimensional simplicesin (cid:98) T i ( m ) , the line segments and the triangles between 0-dimensional simplices are also simplicesin (cid:98) T i ( m ) . It is simple to see that (cid:98) T ( m ) is nice; we call (cid:98) T ( m ) an edge parallel triangulation of W k .For the edge parallel triangulation (cid:98) T ( m ) of W k the following facts are easy to verify.22 act 3.9. The diameter of (cid:98) T ( m ) is at most m . Fact 3.10.
We can indicate uniquely a simplex σ (cid:63) i using (cid:100) log ( k ) (cid:101) bits. Then, using m + bits we canindicate uniquely a combination of two lines: (1) one of the · m + lines that are parallel to either ψ i or ψ i ,and (2) one of the m + lines that are parallel to ψ i . This combination uniquely determines a point x ∈ R and we can efficiently check whether this point belongs to σ (cid:63) i or not. Of course this way the points that lieon the rays from to u i have two different representations but we can easily resolve this discrepancy bychoosing as valid only the representation that is lexicographically first. Hence, we can uniquely determineany vertex in (cid:98) T ( m ) with b = (cid:100) log ( k ) (cid:101) + m + bits. Fact 3.11.
Given a set of binary vectors A = { a i } i where a i ∈ {
0, 1 } b , there exists an efficient procedurethat determines whether conv ( A ) is a simplex of (cid:98) T ( m ) or not. Because of Fact 3.10, we can assume that the labeling λ of (cid:98) T ( m ) is given via a circuit L with b = (cid:100) log ( k ) (cid:101) + m + (cid:100) log ( k ) (cid:101) output bits. The input to the circuit L is therepresentation of a potential vertex x in (cid:98) T ( m ) according to Fact 3.10 and the output is the label λ ( x ) ∈ [ k ] of this vertex x . Observe that Fact 3.10 also guarantees that it is easy to check whetheran input a ∈ {
0, 1 } b to the circuit L corresponds to a valid vertex x ∈ (cid:98) T ( m ) . We are now ready todefine the total search problem that is associated with k -Polygon Tucker’s Lemma. k - polygon -T ucker I nput : A circuit L : {
0, 1 } b → {
0, 1 } (cid:100) log ( k ) (cid:101) , with b = (cid:100) log ( k ) (cid:101) + m + utput : One of the following.1. Two binary vectors a , a ∈ {
0, 1 } b such that a , a correspond to vertices x , x on ∂ (cid:98) T ( m ) with x = θ k ( x ) , but L ( a ) (cid:54) = L ( a ) + ( mod k ) .2. Three binary vectors a , a , a ∈ {
0, 1 } b such that the simplex σ = conv ( { a , a , a } ) belongsto (cid:98) T ( m ) and all the labels L ( a ) , L ( a ) , L ( a ) are different from each other.3. Two binary vectors a , a ∈ {
0, 1 } b such that the edge ψ = conv ( { a , a } ) belongs to (cid:98) T ( m ) and labels L ( a ) , L ( a ) are different and they satisfy L ( a ) − L ( a ) (cid:54) = ± ( mod k ) . Lemma 3.12.
It holds that k - polygon -T ucker is in PPA- k [ ] .Proof. This lemma follows from the proof of k -Polygon Tucker’s Lemma that we presentedin Section 3.1.2. We only need to add the description of the circuits S , P that define thegraph G constructed in the proof. Then, using these circuits we reduce k - polygon -T ucker toI mbalance - mod - k [ ] . As we showed in Lemma 3.8, every solution to the resulting instance ofI mbalance - mod - k [ ] corresponds to a solution of k - polygon -T ucker . Hence, k - polygon -T ucker is in PPA- k [ ] .Since the vertices of the graph G correspond to simplices in (cid:98) T ( m ) and the maximum degreeof any node in G is k , we define the circuits S , P to have 3 · b binary inputs and k · ( b ) outputs,hence S : {
0, 1 } b → ( {
0, 1 } b ) k and P : {
0, 1 } b → ( {
0, 1 } b ) k . For the construction of the circuits,both S and P first check whether an input ( a , a , a ) is valid or not. An input ( a , a , a ) is validin the following cases. 23 If all a , a , a are different, then ( a , a , a ) is valid if a , a , a are in lexicographicalorder and the simplex conv ( { x , x , x } ) , where x i is the point in R that corresponds to a i according to Fact 3.10, is a valid simplex of (cid:98) T ( m ) . Observe that we can check this using apolynomial size circuit by Fact 3.11. (cid:46) If { a , a , a } has two different vectors, then ( a , a , a ) is valid if the lexicographically firstis a , the lexicographically second is a = a and the edge conv ( { x , x } ) , where x i is thepoint in R that corresponds to a i according to Fact 3.10, is a valid simplex of (cid:98) T ( m ) . Observethat we can check this using a polynomial size circuit by Fact 3.11. (cid:46) If a = a = a , then ( a , a , a ) is valid if the point x in R that corresponds to a according to Fact 3.10 is a valid vertex of (cid:98) T ( m ) . Observe that we can check this using apolynomial size circuit by Fact 3.11.If input ( a , a , a ) is not valid, then both circuits S and P output ( a , a , a ) concatenatedwith itself k times. On the other hand, if ( a , a , a ) is valid, then both circuits check whetherthis input corresponds to a relevant simplex as we defined in Section 3.1.2. It is easy to see thatchecking relevance can be done efficiently. Again, if ( a , a , a ) is not relevant, both circuits S and P output ( a , a , a ) concatenated with itself k times. Finally, if ( a , a , a ) is valid and relevant,then we define the edges of the corresponding vertex in G as described in Section 3.1.2. From theconstruction of G , it follows that the successors and the predecessors can be computed efficiently.If the vertex in G that corresponds to ( a , a , a ) has less than k incoming or outgoing edges, thenwe repeat the lexicographically last neighboring vertex enough times such that the number of bitsin the output of both S and P is k · ( b ) .Using our analysis in Section 3.1.2, it follows that the above construction of S and P defines areduction from k - polygon -T ucker to I mbalance - mod - k [ ] .We now define the computational problem associated with the k -Polygon Borsuk-UlamTheorem. For this computational problem we need a representation of the continuous function f that is the input to the k -Polygon Borsuk-Ulam Theorem. Following standard techniques in theliterature, we use arithmetic circuits with gates × ζ (multiplication by a constant), + , − , < , min,and max and rational constants to define this function. k - polygon -B orsuk -U lam I nput : An arithmetic circuit C : B → R , an accuracy parameter ε > L .O utput : One of the following.1. A point x ∈ S such that (cid:13)(cid:13) C ( ˜ θ k ( x )) − ˜ θ k ( C ( x )) (cid:13)(cid:13) ≥ η ( ε , k ) : = ε /8 k .2. Two points x , y ∈ B such that (cid:107)C ( x ) − C ( y ) (cid:107) > L (cid:107) x − y (cid:107) .3. A point x ∗ ∈ B such that (cid:107)C ( x ∗ ) (cid:107) ≤ ε .The first type of solution corresponds to a violation of the boundary conditions. Since wecannot compute θ k exactly, we let ˜ θ k denote a function that computes θ k with error at most ξ ( ε , L , k ) : = ε Lk . The second type of solution corresponds to a violation of L -Lipschitz-continuity.Note that the function computed by the circuit C might not be continuous, because of the24omparison gate. Thus we add this extra violation, which ensures that the function is Lipschitz-continuous. This allows us to relate this problem to k - polygon -T ucker (and in particular to showthat it always has a solution). Lemma 3.13.
The problem k - polygon -B orsuk -U lam reduces to the problem k - polygon -T ucker .Therefore, k - polygon -B orsuk -U lam is in PPA- k.Proof.
This result is obtained by following the idea of the proof of Lemma 3.6, and constructinga labeling L that uses the circuit C as a sub-routine to compute the labels of the corresponding k - polygon -T ucker instance. The details are a bit tricky because we have to account for smallerrors in various computations.In more detail, the regular procedure for picking a label at some point is to first compute anapproximate value of its coordinates and then based on this to compute an approximate valueof the function. However, this might introduce bogus boundary condition violations, even if thefunction perfectly satisfies the boundary conditions. We can resolve this as follows. For any vertex x that lies on the boundary, but not between u and u , we first check how far away the labelobtained through the regular procedure is from the label that satisfies the boundary condition.If it happens that the regular procedure is close to outputting the label that does not violate theboundary conditions then we enforce the output of this label and ignore the output of the regularprocedure. Otherwise, we output the label of the regular procedure. Then, we can show that fromany solution of k -Polygon Tucker we can either extract an approximate zero of C , or a violation ofthe boundary conditions of C (in their approximate version), or a violation of Lipschitz continuity.We will use the triangulation (cid:98) T ( m ) where m is picked sufficiently small so that the diameter ofthe triangulation when mapped to B by the homeomorphism is at most δ = ε Lk . The Booleancircuit computing L performs the following operations. Recall that the input to the circuit is thebits representing the index of the vertex x ∈ W k of the triangulation, as per Fact 3.10.1. Let y ∈ B denote the image of x under the homeomorphism used in the proof of Lemma 3.6.Compute an approximation ˜ y of y with error at most η = δ = ε Lk , i.e., (cid:107) y − ˜ y (cid:107) ≤ η . Notethis is done in two steps, first compute an approximate value for the coordinates of x , whichis given through its index, and compute an approximate value of the homeomorphism giventhe approximate value of the coordinates of x . These steps can be done efficiently by usingstandard techniques [Brent, 1976] so that the total approximation error is η .2. Compute C ( ˜ y ) exactly.3. For each i ∈ [ k ] , compute an estimate v i ( ˜ y ) of the inner product (cid:104)C ( ˜ y ) , u i (cid:105) with error at most η = ε /64 k , where the approximation error is introduced in computing the coordinates of u i .4. If x / ∈ ∂ (cid:98) T ( m ) or if x ∈ conv ( { u , u } ) \ { u } , then output the label argmax i v i ( ˜ y ) (break tiesarbitrarily but deterministically, e.g., lexicographically).5. Otherwise, there exists (cid:96) ∈ [ k − ] and a vertex x (cid:48) ∈ conv ( { u , u } ) \ { u } , such that x = θ (cid:96) k ( x (cid:48) ) . In that case, compute r = max i v i ( ˜ y ) and j = L ( x (cid:48) ) + (cid:96) ( mod k ) . Let η = ε k ( + k ) .If r − v j ( ˜ y ) ≤ η then output L ( x (cid:48) ) + (cid:96) ( mod k ) , otherwise output argmax i v i ( ˜ y ) (againbreak ties arbitrarily but deterministically, e.g., lexicographically).25e will use the following useful invariant: if vertex x has label j , then (cid:104)C ( ˜ y ) , u j (cid:105) ≥ max i (cid:104)C ( ˜ y ) , u i (cid:105) − η − η .This follows from our labeling procedure.We begin by considering standard solutions of k - polygon -T ucker and distinguish betweenthe cases k = k >
3. The other type of solution, namely boundary violations are treated forany k ≥ k = In this case a standard solution to L consists of a simplex with vertices x , x , x thathave labels 1, 2, 3 respectively. Let ˜ y , ˜ y , ˜ y denote the corresponding points computed in theaforementioned step 1. of the computation of L with input x , x , x respectively. Assume that (cid:107)C ( ˜ y i ) (cid:107) ≥ ε for i =
1, 2, 3, since we have found a solution otherwise. Using elementary linearalgebra, it is easy to show that for any ˜ y ∈ B with (cid:107)C ( ˜ y ) (cid:107) ≥ ε , it holds that max i (cid:104)C ( ˜ y ) , u i (cid:105) ≥ ε /2and min i (cid:104)C ( ˜ y ) , u i (cid:105) ≤ − ε /2.Let j = argmin i (cid:104)C ( ˜ y ) , u i (cid:105) . Then it follows that (cid:104)C ( ˜ y ) , u j (cid:105) ≤ − ε /2, which implies that (cid:104)C ( ˜ y j ) , u j (cid:105) ≤ − ε /4, since |(cid:104)C ( ˜ y ) , u j (cid:105) − (cid:104)C ( ˜ y j ) , u j (cid:105)| ≤ (cid:107)C ( ˜ y ) − C ( ˜ y j ) (cid:107) ≤ L (cid:107) ˜ y − ˜ y j (cid:107) ≤ L ( δ + η ) ≤ ε /4unless ˜ y and ˜ y j yield a violation of L -Lipschitz continuity of C , in which case we have founda solution. Now, recall that x j has label j . By the invariant, it must hold that (cid:104)C ( ˜ y j ) , u j (cid:105) ≥ max i (cid:104)C ( ˜ y j ) , u i (cid:105) − η − η . But since max i (cid:104)C ( ˜ y j ) , u i (cid:105) ≥ ε /2, this means that (cid:104)C ( ˜ y j ) , u j (cid:105) ≥ ε /2 − η − η , which contradicts the fact that (cid:104)C ( ˜ y j ) , u j (cid:105) ≤ − ε /4. k > In this case a solution to L consists of an edge x , x of the triangulation such that the twovertices have labels j , j that differ by more than one. Assume that (cid:107)C ( ˜ y ) (cid:107) ≥ ε and (cid:107)C ( ˜ y ) (cid:107) ≥ ε .Note that ˜ y lies in some cone conv ( { , u t , u t + } ) . By elementary linear algebra, one can showthat max i ∈{ t , t + } (cid:104)C ( ˜ y ) , u i (cid:105) − max i / ∈{ t , t + } (cid:104)C ( ˜ y ) , u i (cid:105) ≥ ε ( − cos ( π / k )) ≥ ε k (1)Since ε k ≥ η + η , it follows that the label j of x must be one of t or t +
1. As a result, j / ∈ { t , t + } , since it cannot be adjacent to j . Furthermore, since j was picked as the label of x , by the invariant it holds that (cid:104)C ( ˜ y ) , u j (cid:105) ≥ max i (cid:104)C ( ˜ y ) , u i (cid:105) − η − η . Together with equation(1), it follows that (cid:104)C ( ˜ y ) , u j (cid:105) − (cid:104)C ( ˜ y ) , u j (cid:105) ≥ ε k − η − η .On the other hand, since x has label j , this means that (cid:104)C ( ˜ y ) , u j (cid:105) − (cid:104)C ( ˜ y ) , u j (cid:105) ≤ η + η .As a result, it must be that (cid:107)C ( ˜ y ) − C ( ˜ y ) (cid:107) ≥ ε k − η − η . Note that this is a violation ofLipschitz continuity, since normally we should have (cid:107)C ( ˜ y ) − C ( ˜ y ) (cid:107) ≤ L (cid:107) ˜ y − ˜ y (cid:107) ≤ L ( δ + η ) and this quantity is smaller than ε k − η − η .26 oundary violations. Let x be a violation of the boundary conditions, i.e., the label at vertex x is j , and the label at θ k ( x ) is j (cid:54) = j + ( mod k ) . This means that at least one of the twovertices did not obtain its “intended” label. Without loss of generality, assume that x has notobtained its intended label. Let (cid:96) ∈ [ k − ] and x ∈ conv ( { u , u } ) \ { u } be such that θ (cid:96) k ( x ) = x .Since x did not obtain its “intended” label, it follows that max i v i ( ˜ y ) − v j ( ˜ y ) > η , where ˜ y is thecorresponding point in B computed in step 1. This implies thatmax i (cid:104)C ( ˜ y ) , u i (cid:105) − (cid:104)C ( ˜ y ) , u j (cid:105) > η − η .On the other hand, since x ∈ conv ( { u , u } ) \ { u } has label j − (cid:96) ( mod k ) , it holds thatmax i (cid:104)C ( ˜ y ) , u i (cid:105) − (cid:104)C ( ˜ y ) , u j − (cid:96) ( mod k ) (cid:105) ≤ η .where ˜ y is the corresponding point of x in B computed in step 1. By noting thatmax i (cid:104)C ( ˜ y ) , u i (cid:105) − (cid:104)C ( ˜ y ) , u j − (cid:96) ( mod k ) (cid:105) = max i (cid:104) θ (cid:96) k ( C ( ˜ y )) , u i (cid:105) − (cid:104) θ (cid:96) k ( C ( ˜ y )) , u j (cid:105) we thus obtain that there exists i ∈ [ k ] such that |(cid:104)C ( ˜ y ) , u i (cid:105) − (cid:104) θ (cid:96) k ( C ( ˜ y )) , u i (cid:105)| > ( η − η ) /2.This implies that (cid:107)C ( ˜ y ) − θ (cid:96) k ( C ( ˜ y )) (cid:107) ≥ ( η − η ) /2. Since C ( ˜ y ) − θ (cid:96) k ( C ( ˜ y )) = (cid:96) − ∑ i = θ ik C ( θ − ik ( ˜ y )) − θ i + k ( C ( θ (cid:96) − i − k ( ˜ y ))) it follows that there exists i ∈ {
0, 1, . . . , (cid:96) − } such that (cid:107) θ ik C ( θ − ik ( ˜ y )) − θ i + k ( C ( θ (cid:96) − i − k ( ˜ y ))) (cid:107) ≥ ( η − η ) / ( k ) .Now let x ∗ = θ (cid:96) − i − k ( x ) . Then ˜ y ∗ yields a violation of the boundary conditions for C by notingthat (cid:107)C ( ˜ θ k ( ˜ y ∗ )) − ˜ θ k ( C ( ˜ y ∗ )) (cid:107) ≥ ( η − η ) / ( k ) − (cid:107)C ( ˜ θ k ( ˜ y ∗ )) − C ( θ k ( ˜ y ∗ )) (cid:107) − (cid:107) ˜ θ k ( C ( ˜ y ∗ )) − θ k ( C ( ˜ y ∗ )) (cid:107)≥ ( η − η ) / ( k ) − L ( η + ξ ( ε , L , k )) − ξ ( ε , L , k ) unless we find a violation of Lipschitz continuity. Indeed, note that ( η − η ) / ( k ) − L ( η + ξ ( ε , L , k )) − ξ ( ε , L , k ) ≥ η ( ε , k ) . k -Polygon Borsuk-Ulam and k -Polygon Tucker are PPA- k [ ] -hard In this section we show the PPA- k [ ] -hardness of both k - polygon -T ucker and k - polygon -B orsuk -U lam . We start by observing that using the ideas from Section 3.1.1 we can prove that k - polygon -T ucker is reducible to k - polygon -B orsuk -U lam . Then we show that k - polygon -T ucker isPPA- k [ ] -hard. Finally the results of this section together with the results of the previousSection 3.1.3 imply that both k - polygon -T ucker and k - polygon -B orsuk -U lam are PPA- k [ ] -complete. Lemma 3.14.
The problem k - polygon -T ucker reduces to the problem k - polygon -B orsuk -U lam . roof. This Lemma follows from the proof of Lemma 3.5 combined with standard arguments onhow to construct an arithmetic circuit approximately computing the piecewise linear function g : B → R described in Lemma 3.5. Recall that the function g is constructed by interpolatingwithin simplices of the triangulation using the labels given by L . Given a precision parameter γ and the Boolean circuit L , we can construct an arithmetic circuit C that computes the function g with error at most γ , using standard techniques [Etessami and Yannakakis, 2010, Daskalakisand Papadimitriou, 2011, Goldberg and Hollender, 2019]. To be more precise, we can invokeTheorem E.2 from [Fearnley et al., 2020] by observing that the function g is polynomially approxi-mately computable via standard numerical analysis techniques Brent [1976]. Furthermore, thefunction computed by C will be Lipschitz-continuous with a Lipschitz constant ˜ L that has bit-sizepolynomial in m and log ( γ ) (where m is the parameter of the triangulation used by L ).For any x ∈ B that does not lie in a simplex that is a solution of L , it must hold that (cid:107) g ( x ) (cid:107) ≥ − γ > ε , any point with (cid:107)C ( x ) (cid:107) ≤ ε will yield a solution of the k - polygon -T ucker instance.It remains to show that no bogus boundary condition violations occur. Note that unlessthe simplex containing x ∈ ∂ B yields a boundary condition violation for L , it must hold that g ( θ k ( x )) = θ k ( g ( x )) (as shown in Lemma 3.5). Thus, it follows that (cid:107)C ( ˜ θ k ( x )) − ˜ θ k ( C ( x )) (cid:107) ≤ (cid:107)C ( θ k ( x )) − θ k ( C ( x )) (cid:107) + (cid:107)C ( ˜ θ k ( x )) − C ( θ k ( x )) (cid:107) + (cid:107) ˜ θ k ( C ( x )) − θ k ( C ( x )) (cid:107)≤ γ + ˜ L · ξ ( ε , L , k ) + ξ ( ε , L , k ) .Note that we want this quantity to be strictly less than η ( ε , k ) . Thus, we pick ε = γ and L so that• L ≥ ˜ L ,• γ < − ε ,• 2 γ + ˜ L · ξ ( ε , L , k ) + ξ ( ε , L , k ) < η ( ε , k ) ,which can easily be achieved by picking γ sufficiently small and L sufficiently large. Note thatsince C is ˜ L -Lipschitz-continuous, it will also be L -Lipschitz-continuous.Now we present our main proposition in this section. Proposition 3.15.
For all k ≥ , k - polygon -T ucker is PPA- k [ ] - hard . Recall that PPA- k [ ] = ∩ p ∈ PF ( k ) PPA- p , where PF ( k ) denotes the set of prime factors of k . Proof.
To prove this hardness result we reduce from B ipartite - mod - k [ ] (see Section 2.2 for theformal definition). Recall that in this problem we are given a bipartite graph on the set of nodes A ∪ B , where A = { } × {
0, 1 } n and B = { } × {
0, 1 } n , such that the node 00 n ∈ A has degree1. The goal is to find any other node that has degree (cid:54) = k . All nodes have degree in {
0, 1, . . . , k } and we can assume (see e.g., [Hollender, 2019, Section 4.2]) that the circuit C whichimplicitly represents the bipartite graph is consistent, i.e., for all x , y we have y ∈ C ( x ) iff x ∈ C ( y ) .Consider the regular k -polygon W k in R with the edge parallel triangulation (Definition 8).For any i ∈ Z k let R ( i , i + ) denote the triangle with endpoints { , u i , u i + } . In this proof we referto nodes of the B ipartite - mod - k [ ] instance and to vertices of the triangulation of W k .28o every node x ∈ ( A \ { n } ) ∪ B we associate a distinct interval on the outer boundary of R (
1, 2 ) , i.e., on the edge conv ( { u , u } ) . This interval, which we denote by K ( x ) , is picked suchthat it covers 4 k vertices of the triangulation. It is easy to see that we can pick the triangulation tobe fine enough so that there are indeed enough vertices on conv ( { u , u } ) to associate a distinctinterval of 4 k vertices to each node x ∈ ( A \ { n } ) ∪ B . Since the triangulation is symmetric onthe boundary with respect to θ k , we can immediately also extend this association to the outerboundary of R ( i , i + ) for all other i . In other words, for any i ∈ Z k and any x ∈ ( A \ { n } ) ∪ B ,we let K i ( x ) = ( θ k ) i − ( K ( x )) , which is an interval of 4 k vertices on conv ( { u i , u i + } ) . Thus, for any x there are k distinct intervals on the boundary associated to it, one in each of conv ( { u i , u i + } ) , i ∈ Z k .In the rest of this proof we explain how to assign a label in Z k to every vertex of thetriangulation so that the k - polygon -T ucker boundary conditions are satisfied and any solution(i.e., a trichromatic triangle or an edge with non-consecutive labels) must contain a vertex that liesin some interval K i ( x ) where x is a solution-node of B ipartite - mod - k [ ] (i.e., a node with degree (cid:54) = k ). This will ensure that from any solution of the k - polygon -T ucker instance we caneasily obtain a solution-node of the B ipartite - mod - k [ ] instance.We begin by defining the “environment” label for every vertex of the triangulation. Thiscorresponds to the standard label that the vertex will have, unless we specify it otherwise in theconstruction. Any vertex lying in R ( i , i + ) \ conv ( { , u i + } ) has the environment label i . Next,we define “cables”, which will be used to embed the edges of the B ipartite - mod - k [ ] instance inour construction. Wires and Cables.
A “wire” has an associated label i ∈ Z k and it simply consists of a path ofvertices in the triangulation such that all vertices on the path have the label i . A “cable” is madeout of k − Z k \ { i } , then the wires are arranged according to their labels inthe order i + i +
2, . . . , i + ( k − ) from right to left, in the forward direction of the cable. Notethat while wires are not directed, we can define a direction for every cable, based on the order ofthe wires inside the cable. A cable using the labels Z k \ { i } is only allowed to exist inside a regionwith environment label i . This ensures that any vertex that is adjacent to either side of the cable,and thus to the wire labeled i + i + ( k − ) , is labeled i and does not introduce a solution.The construction of the cables ensures that the wires are “isolated” from each other and from theenvironment, in the sense that no solution is introduced along the cable. However, if the start orend of a cable using the labels Z k \ { i } is allowed to “touch” the environment label i , then thiswill necessarily yield a trichromatic triangle at that point. See Figure 3 for an illustration of how acable is constructed.It is easy to see that a cable can turn without introducing solutions. Next, let us see how a cablecan transition from one environment to another. Consider a cable in environment i ∈ Z k , i.e., ituses the labels Z k \ { i } and lies in R ( i , i + ) . If the cable arrives on the boundary conv ( { , u i + } ) of R ( i , i + ) , we can transform it into a cable that uses the labels Z k \ { i + } and continues into R ( i + i + ) , i.e., in the environment i +
1, on the other side of conv ( { , u i + } ) . Importantly,the direction of the cable does not change and we do not introduce any new solutions. Thistransformation of the cable is shown in Figure 4. The idea is simple. Consider a cable inenvironment i that arrives on conv ( { , u i + } ) moving forward. The wires are arranged accordingto their labels in the order i + i +
2, . . . , i + ( k − ) from right to left, in the forward direction of29 Figure 3: A cable for the case k =
5. The cable uses the labels Z \ { } and the environmenthas label 1. The labels are color-coded as indicated on the left-hand side, i.e., green is label 1,yellow is label 2 etc. The forward direction of the cable is indicated by an arrow on the right-handside. Note that the portion of the cable shown in the figure does not introduce any solution of5- polygon -T ucker .the cable. When the cable reaches conv ( { , u i + } ) , the right-most wire (which is labeled i +
1) isdropped from the cable, i.e., it merges into the environment i + R ( i + i + ) . On the otherside of the cable, a new wire with label i = i + + ( k − ) is created by using the environment i of R ( i , i + ) . Thus, we obtain a cable with wires i +
2, . . . , i + + ( k − ) (from right to left) in theenvironment i +
1, as desired. It is easy to see that this construction does not introduce any newsolutions, because we have ensured that non-consecutive labels do not “touch”.When a cable starts or ends on the outer boundary of R ( i , i + ) , i.e., on conv ( { u i , u i + } ) , the k - polygon -T ucker boundary conditions force a cable to start or end at the corresponding positionin each of the regions R ( j , j + ) , j ∈ Z k \ { i } . These k − R ( i , i + ) ends on the outer boundary,then the k − k − Construction of the instance.
Before we begin, let us introduce the following useful terminology.Every node x of the B ipartite - mod - k [ ] instance has some number (cid:96) ∈ {
0, 1, . . . , k } of neighbors,as given by C ( x ) . For any i ∈ [ (cid:96) ] , we define the “ i th neighbor of x ” to be the i th node in thelexicographically ordered list of neighbors of x .We are now ready to begin describing the instance we construct. Recall that we have definedan environment label for every vertex of the triangulation except . Now consider the labelingwhere every vertex is simply labeled by its environment label. Clearly, this labeling satisfies the k - polygon -T ucker boundary conditions. Furthermore, no matter how we pick the label of , therewill be a solution there, since is adjacent to all environments. Now it is easy to see that we can“move” this solution by locally modifying some of the labels. Namely, instead of having all labelsmeet at , we can instead construct a cable that uses the labels Z k \ { } and moves into the regionwith environment label 1. As a result, there is no longer a solution at , but instead there is now asolution at the end of the cable. In more detail, every environment label except 1 yields a wirewith the corresponding label and the wires are arranged into a cable that uses the labels Z k \ { } .Figure 5 illustrates this construction in the case k = n ∈ A has exactly one neighbor y ∈ B . Let j ∈ [ k ] be the number suchthat 00 n is the j th neighbor of y. The cable that we just constructed at the center of the instance30 u R (1 , R (2 , Figure 4: Transformation of a cable for the case k =
5. In the region R (
1, 2 ) the cable uses labels Z \ { } and has environment 1. When the cable reaches region R (
2, 3 ) , the construction shownin the figure ensures that from now on, the cable uses labels Z \ { } and has environment 2. Thecable uses the labels Z \ { } and the environment has label 2. Note that the transformation ofthe cable as shown in the figure does not introduce any solution of 5- polygon -T ucker .will be routed so that it ends at K j ( y ) (a segment on the outer boundary of R ( j , j + ) , as definedabove). Furthermore, for any x ∈ A \ { n } and y ∈ B such that x and y are neighbors, there willbe a cable starting at K i ( x ) and ending at K j ( y ) , where i , j ∈ [ k ] are such that x is the j th neighborof y , and y is the i th neighbor of x . This ensures that the following properties hold.Consider a node x ∈ A \ { n } that has (cid:96) neighbors, where (cid:96) ∈ {
0, 1, . . . , k } . If (cid:96) =
0, i.e., x is an isolated node, then for all i ∈ [ k ] there is no cable at K i ( x ) . In particular, there is no k - polygon -T ucker solution in any K i ( x ) . If (cid:96) ∈ [ k ] , then for all i ∈ [ (cid:96) ] there is a cable starting at K i ( x ) (and ending at some K j ( y ) ). For all i ∈ [ k ] \ [ (cid:96) ] , there is a start of a cable at K i ( x ) , but thecable just stops immediately and does not go anywhere. This ensures that the k - polygon -T ucker boundary conditions are satisfied, but also that there is a k - polygon -T ucker solution at K i ( x ) forall i ∈ [ k ] \ [ (cid:96) ] (because of an exposed end of cable). Thus, we obtain that• if (cid:96) ∈ [ k − ] , there is a solution at K i ( x ) for some i ∈ [ k ] ,• if (cid:96) ∈ { k } , then there is no solution in any K i ( x ) .Similarly, consider a node y ∈ B that has (cid:96) neighbors, where (cid:96) ∈ {
0, 1, . . . , k } . If (cid:96) =
0, i.e., y is an isolated node, then for all j ∈ [ k ] there is no cable at K j ( y ) . In particular, there is no k - polygon -T ucker solution in any K j ( y ) . If (cid:96) ∈ [ k ] , then for all j ∈ [ (cid:96) ] there is a cable endingat K j ( y ) (that started at some K i ( x ) ). For all j ∈ [ k ] \ [ (cid:96) ] , there is an end of a cable at K j ( y ) ,but the cable just started there and does not come from anywhere else. This ensures that the k - polygon -T ucker boundary conditions are satisfied, but also that there is a k - polygon -T ucker solution at K j ( y ) for all j ∈ [ k ] \ [ (cid:96) ] (because of an exposed start of cable). Thus, we obtain that31 orward directionof the cable u u u u u Figure 5: Construction around the origin for the case k =
5. Even though all the environments“meet” at the origin, the construction shown in the figure ensures that there is no solution aroundthe origin, but instead a cable is created. The labels are color-coded as in Figure 3 and Figure 4.• if (cid:96) ∈ [ k − ] , there is a solution at K j ( y ) for some j ∈ [ k ] ,• if (cid:96) ∈ { k } , then there is no solution in any K j ( y ) .As a result, if the cables can indeed be constructed to connect the various K i ( x ) and K j ( y ) asdesired, then any k - polygon -T ucker solution of the instance will have to be next to some K i ( x ) such that x is a solution-node or next to some K j ( y ) such that y is a solution node. One immediateobstacle to routing the cables as desired is that we are working in two dimensions and it is verylikely that cables will have to cross each other. Fortunately, there is a simple “trick” that has beenused in prior work to resolve this issue [Chen and Deng, 2009]. Consider the following idea:cut the two cables that want to cross each other at the point of crossing. This creates two endsof cables and two starts of cables. It is easy to see that we can connect an end of cable with astart of cable, and the other end with the other start, so that no crossing occurs anymore. Thismodification of the cables is completely local and does not have any impact on the rest of theinstance. Constructing the labeling function.
Since we want to be able to construct a circuit for thelabeling function, we need to be a bit more precise about the path followed by every cable. Oneway to achieve this is to reserve a separate “circular lane” for each pair ( y , j ) , where y ∈ B and j ∈ [ k ] . A circular lane is a path of sufficient width that simply stays parallel to the outer boundary32n each region R ( i , i + ) and thus makes a full “circle” around the center of the domain. Bypicking a fine enough triangulation, we can ensure that there is a separate, disjoint circular lane L y , j for each pair ( y , j ) , where y ∈ B and j ∈ [ k ] . Then the cable going from some K i ( x ) to some K j ( y ) will be routed as follows. Starting at K i ( x ) , move perpendicularly to the outer boundarytowards the inside of the domain, until the circular lane L y , j is reached. Next, follow the lanein clockwise direction. When following the lane, the cable might have to transition from someenvironment to the next and this is implemented as described earlier. The cable stops followingthe lane when it reaches the part of the lane lying just “above” K j ( y ) . In other words, the cablestops following the lane when it is at the point where it can just turn left and move straighttowards the boundary to end up at K j ( y ) , as desired. Clearly, we can pick the triangulation fineenough so that this routing is indeed well defined.This construction has two advantages. First of all, it ensures that any crossing involves at mosttwo cables, not more. We can then use the trick described above to locally resolve these crossings.Furthermore, it ensures that we can construct a Boolean circuit computing the labeling function.Indeed, given any vertex of the triangulation we can easily determine on which circular lane andon which path perpendicular to the outer boundary it lies. This gives us enough information tothen use the B ipartite - mod - k [ ] circuit C as a sub-routine to determine whether the vertex lieson a cable and if so, how exactly the cable behaves locally (including possible crossing-avoidingtrick). The circuit C only needs to be queried a constant number of times and thus the resultingcircuit for the labeling will have polynomial size with respect to the size of C and n . We omit thefull details, since this part of the proof is essentially the same as in prior work (see e.g., [Chen andDeng, 2009]).Based on the results presented in this and the previous section we get the following theorem. Theorem 3.16.
The problems k - polygon -T ucker and k - polygon -B orsuk -U lam are both PPA- k [ ] -complete. In particular, if k = p r is a prime power, then k - polygon -T ucker and k - polygon -B orsuk -U lam are PPA- p -complete. p -complete The main result of this section is the PPA- p -completeness of the computational problem associatedwith the BSS Theorem. We refer to this problem as p -BSS and we define it formally in Section 4.3.We state the main theorem of the section below. Theorem 4.1.
For every prime p, the problem p -BSS is PPA- p-complete.
In order to prove the theorem, first we show that the BSS theorem is equivalent to a generalizationof Tucker’s Lemma, which we call BSS-T ucker , and then we define the corresponding compu-tational problems and show that they are equivalent. Then, we show the PPA- p -completenessof BSS-T ucker , which then implies Theorem 4.1. We show the PPA- p -hardness via a reductionfrom the p - polygon -T ucker problem, proven to be PPA- p -complete in the previous section. Themembership in PPA- p is proven in Section 5, where we show that BSS-T ucker reduces to anothervariant of Tucker’s lemma, which we call Z p - star -T ucker (Proposition 5.4), and for which weprove membership in PPA- p . 33 .1 The BSS Theorem and Equivalent Formulations Our notation follows that of Bárány et al. [1981]. Let p ≥ n ∈ N and let P = { ( v , v , . . . , v p ) ∈ ( R n ) p : ∑ pi = v i = } and P = { u ∈ P : (cid:107) u (cid:107) ≤ } . It holds that P ∼ = R n ( p − ) and P ∼ = B n ( p − ) .Note that P is a hyperplane of R np with dimension n ( p − ) . Let B be an orthogonal basis of P in R np and let φ : P → B n ( p − ) be the function that maps x ∈ P to its coefficients in base B .Then, φ is a homeomorphism of P and B n ( p − ) . Notice that φ ( ∂ P ) = S n ( p − ) − and that φ and φ − are efficiently computable. The same mapping shows that P and R n ( p − ) are homeomorphic.Let θ ( v , . . . , v p ) = ( v p , v , . . . , v p − , v p − ) .The function θ has order p ; namely the composition by itself p times is equal to the identity.Also, for all i ∈ {
1, . . . , p − } and all x ∈ P \ { } , θ i ( x ) (cid:54) = x . In other words, for any i < p , θ i restricted to P \ { } has no fixed points. Hence, θ acts freely on P \ { } . It follows with a similarargument that the function θ acts freely also on P \ { } and on ∂ P .The original statement of the BSS Theorem requires the notion of CW-complexes, but sincein our work we do not use CW-complexes, we will not define them formally. Intuitively, aCW-complex consists of building blocks that can be topologically glued together.Let p be a prime, n ≥ X be a CW-complex consisting of p copies of the n ( p − ) -dimensional ball glued on their boundaries.Let α : R n ( p − ) → R n ( p − ) such that α = φ ◦ θ ◦ φ − . Note that α acts freely on R n ( p − ) \ { } , B n ( p − ) \ { } and S n ( p − ) − . Let ω be the extension of α on X defined as follows: ω ( y , r , q ) = ( α y , r , q + ( mod p )) ,where ( y , r , q ) denotes the point of the q -th ball with radius r and direction y ∈ S n ( p − ) − .The map ω is a free action on X and the following theorem holds: Theorem 4.2 (BSS Theorem, [Bárány, Shlosman, and Szücs, 1981]) . For the mapping ω and anycontinuous map h : X → R n , there exists an x ∈ X such that h ( x ) = h ( ω x ) = · · · = h ( ω p − x ) . The following equivalent formulations of Theorem 4.2 are useful for defining the computationalproblem related to BSS.
Theorem 4.3 (BSS Theorem, equivalent formulations) . The following statements are equivalent to theBSS Theorem:1. Let g : P → P be continuous and such that g ( θ x ) = θ g ( x ) for all x ∈ ∂ P . Then, there exists x ∈ P such that g ( x ) = .2. Let g : B n ( p − ) → R n ( p − ) be continuous and such that g ( α x ) = α g ( x ) for all x ∈ S n ( p − ) − . Then,there exists x ∈ B n ( p − ) such that g ( x ) = .Proof. We first show that the two statements are equivalent and then that statement (2) is equivalentto Theorem 4.2. (1) ⇐⇒ (2) The equivalence follows from P ∼ = B n ( p − ) and P ∼ = R n ( p − ) , as well as from theequivariance of φ . 34 ⇐⇒ (BSS) We first show that the BSS Theorem implies (2). Recall that the CW-complex X consists of p copies of B n ( p − ) with their boundaries “glued” together. Define h : X → R n as follows: for x = ( y , r , i ) ∈ X , let h ( x ) = [ φ − ◦ g ◦ α − i ( r y )] − i ∈ R n , where for j ∈ Z p ≡ [ p ] , [ · ] j denotes the j -th component of an element in ( R n ) p (i.e., [( v , . . . , v p )] j = v j ). The mapping h is well-defined and continuous on the glued boundary, because for all i and y ∈ S n ( p − ) − ,we have h ( y , 1, i ) = [ φ − ◦ g ◦ α − i ( y )] − i = [ θ − i φ − ◦ g ( y )] − i = [ φ − ◦ g ( y )] (which does notdepend on i ). Finally, note that any x = ( y , r , 1 ) ∈ X with h ( x ) = h ( ω x ) = · · · = h ( ω p − x ) yields z = r y ∈ B n ( p − ) with [ φ − ◦ g ( z )] = · · · = [ φ − ◦ g ( z )] p , which implies φ − ◦ g ( z ) = P ), and thus g ( z ) = B n ( p − ) with the first ball in the CW-complex X .Given a continuous function h : X → R n , define g : B n ( p − ) → R n ( p − ) by g ( x ) = φ ( h ( ω p x ) − h ( ω p − x ) , h ( ω p − x ) − h ( ω p − x ) , . . . , h ( ω x ) − h ( x )) . It is easy to check that g ( α x ) = α g ( x ) for all x ∈ S n ( p − ) − by noting that ω x = α x for such x . In this section, we define a generalization of Tucker’s Lemma, that we call the BSS-T ucker
Lemma and show that it is equivalent to the BSS Theorem. The BSS-T ucker
Lemma applies totriangulations of P that have some special properties.For j ∈ [ n ] , let e j ∈ R n be the j -th unit vector. For ( i , j ) ∈ Z p × [ n ] , let e i , j = ( p − ) ( − e j , . . . , − e j , ( p − ) e j , − e j , . . . , − e j ) ∈ P where the term ( p − ) e j is in the i -th position. For each s = ( s , . . . , s n ) ∈ [ p ] n , we define thesimplex σ s = { } ∪ { e i , j s.t. for each j ∈ [ n ] , i ∈ Z p \ { s j }} . Note that | V ( σ s ) | = ( p − ) n + Lemma 4.4. T ∗ = { τ : τ ⊆ σ s with s ∈ [ p ] n } is a triangulation of P .Proof. Let C = conv (cid:16) ( e i , j ) ( i , j ) ∈ Z p × [ n ] (cid:17) . Then, by definition of T ∗ , C ∼ = (cid:107) T ∗ (cid:107) . So, the lemma followsfrom the fact that C ∼ = P . Definition 9.
We say that a triangulation T of P is nice if it satisfies the following two conditions:• If σ ∈ T ∩ ∂ P then θσ ∈ T .• T refines the triangulation T ∗ (that is, for each σ ∈ T there is τ ∈ T ∗ with σ ⊆ τ ). Theorem 4.5 (BSS-T ucker
Lemma) . Let T be a nice triangulation of P . Let λ = ( λ , λ ) : V ( T ) → Z p × [ n ] be a labeling such that for all x ∈ ∂ T, λ ( θ x ) = ( λ ( x ) + λ ( x )) . Then, there exists a ( p − ) -simplex σ of T such that λ ( σ ) = Z p × { j } for some j ∈ [ n ] , where λ ( σ ) = { λ ( x ) : x ∈ V ( σ ) } . Lemma 4.6.
The BSS Theorem (Theorem 4.3) implies the
BSS-T ucker
Lemma (Theorem 4.5).Proof.
We interpret each label ( i , j ) ∈ Z p × [ n ] as the vector e i , j and we set g to be the extensionof λ to a piecewise linear function on P . Notice that g ( θ x ) = θ g ( x ) for x ∈ ∂ P . Then, it followsfrom Theorem 4.3 that there exists an x ∈ P such that g ( x ) =
0. The lemma follows by noting thatany convex combination of different vectors e i , j that equals must contain p vectors e i , j , . . . , e i p , j p such that { i k , j k } k ∈ [ p ] = Z p × { j } . Hence, the point x lies in a simplex with a ( p − ) -dimensionalface σ such that λ ( σ ) = Z p × { j } for some j ∈ [ n ] .35 emma 4.7. The
BSS-T ucker
Lemma (Theorem 4.5) implies the BSS Theorem (Theorem 4.3).Proof.
We show that BSS-T ucker implies Statement (1) of Theorem 4.3. Using standard arguments,it suffices to show that for every ε > x such that (cid:107) g ( x ) (cid:107) ∞ ≤ ε . Since g : P → P is a continuous function in a compact set, it is also uniformly continuous. Thus, forevery ε >
0, there exists δ such that if (cid:107) x − x (cid:48) (cid:107) < δ , then (cid:107) g ( x ) − g ( x (cid:48) ) (cid:107) ∞ < ε / n . Assume that T is a nice triangulation of P with diameter at most δ .For any point x ∈ V ( T ) , the label λ ( x ) = ( i ∗ , j ∗ ) is defined as follows. For ( i , j ) ∈ Z × [ n ] ,let [ g ( x )] i , j denote the ( i , j ) -coordinate of g ( x ) ∈ P . First, consider the case where g ( x ) (cid:54) = .Then, pick j ∗ = argmax j ∈ [ n ] max i ∈ [ p ] [ g ( x )] i , j . Break ties by picking the smallest such j . Let S = { i : [ g ( x )] i , j ∗ = max (cid:96) [ g ( x )] (cid:96) , j ∗ } and pick i ∗ = T p ( S ) , where T p is the Z p -equivariant tie-breaking function of Definition 19. Note that S / ∈ { ∅ , [ p ] } , because g ( x ) (cid:54) = and by the choice of j ∗ . If g ( x ) = , then if x = , we assign an arbitrary label, and if x (cid:54) = , we use the same procedureas above but with x instead of g ( x ) .With this definition of the labeling, it is easy to check that for any x ∈ ∂ T , we alwayshave λ ( θ x ) = ( λ ( x ) + λ ( x )) , since g ( θ x ) = θ g ( x ) . Hence, by Theorem 4.5 there exists a σ = { x , . . . , x p } ∈ T and a j ∗ ∈ [ n ] such that λ ( x i ) = ( i , j ∗ ) for all i ∈ [ p ] .The lemma follows by showing that there exists i ∈ [ p ] such that (cid:107) g ( x i ) (cid:107) ∞ ≤ ε . First of all,note that for any x ∈ P , by definition we have that (cid:107) g ( x ) (cid:107) ∞ ≤ n · max i , j [ g ( x )] i , j . Now, assume forthe sake of contradiction that (cid:107) g ( x i ) (cid:107) ∞ > ε for all i ∈ [ p ] . Then, it follows that [ g ( x i )] i , j ∗ > ε / n forall i ∈ [ p ] . But by the choice of diameter for the triangulation and uniform continuity of g , thisimplies that [ g ( x )] i , j ∗ > i ∈ [ p ] , which contradicts x ∈ P . p -BSS-Tucker and p -BSS Motivated by Theorem 4.3 and Theorem 4.5, we define the computational problems correspondingto the BSS Theorem and the BSS-T ucker
Lemma. Combining the proof ideas in Lemma 4.6 andLemma 4.7, together with some efficiency requirements of the triangulation, as per Definition 17,we show that the two computational problems are polynomially equivalent.As before, we assume that the input functions are represented as arithmetic circuits withoperations × ζ , + , − , < , min, and max and rational constants. Similarly to our definition of k - polygon -B orsuk -U lam , and for the same reasons, we will add a solution type to ensure that itis Lipschitz-continuous.The computational analogue of the BSS theorem is based on statement (2) of Theorem 4.3. Ittakes as input an arithmetic circuit, which evaluates the function g : B n ( p − ) → R n ( p − ) . In orderto deal with the polynomial-size representation of any of the solutions we allow as a solutionany approximate violation of the equivariance condition of the BSS theorem, as we did with p - polygon -B orsuk -U lam in Section 3. 36 -BSS: I nput : An integer n ≥
1, an accuracy parameter ε >
0, a Lipschitz constant L , and an arithmeticcircuit C .O utput :1. A point x ∈ S n ( p − ) − such that (cid:107)C ( α x ) − α C ( x ) (cid:107) > η ( ε , p ) : = ε /8 p
2. Two points x , y ∈ B n ( p − ) such that (cid:107)C ( x ) − C ( y ) (cid:107) > L (cid:107) x − y (cid:107)
3. A point x ∗ ∈ B n ( p − ) such that (cid:107)C ( x ∗ ) (cid:107) ∞ ≤ ε A valid output of the p -BSS problem is either a point that violates the boundary condition ofTheorem 4.3, two points that violate the Lipschitz-continuity or an approximate root of thefunction g .The computational analogue of the BSS-T ucker Lemma is based on Theorem 4.5 and isparameterized by a “triangulation scheme” T . Namely, given an m ∈ N , T ( m ) yields a nicetriangulation T with diameter at most 1/2 m . The triangulation T is given through two arithmeticcircuits, index and value (see Definition 17), that have size polynomial in n and m . Assume that for m = T yields the index ∗ and value ∗ circuits of T ∗ . p -BSS-Tucker[ T ]: I nput : An arithmetic circuit λ that outputs a number in Z p × [ n ] .O utput :1. A vertex x ∈ ∂ T such that λ ( θ x ) (cid:54) = ( λ ( x ) + λ ( x ))
2. A simplex σ ∗ ∈ T such that λ ( σ ∗ ) = Z p × { j } for some j ∈ [ n ] The valid outputs of the p -BSS-T ucker problem correspond either to points that violate theboundary condition of Theorem 4.5 or to a fully labeled simplex in T . Remark:
To efficiently check whether x ∈ ∂ T , we use the index ∗ and value ∗ circuits of T ∗ , whichexist by assumption on T for m =
0. If value ∗ ( index ∗ ( x )) does not contain , then by definition of T ∗ and the fact that T is a nice triangulation x ∈ ∂ T . Theorem 4.8. p- BSS-T ucker [ T ] and p- BSS are polynomially equivalent.Proof.
First, we show that p -BSS-T ucker [ T ] reduces to p -BSS and then that p -BSS reduces to p -BSS-T ucker [ T ]. For simplicity of the presentation and in order for the main ideas to be clearwe present here a proof sketch of these reductions where we assume that η ( ε , p ) = p -BSSproblem. The complete proof then follows using the tedious but straightforward caseanalysis that we used in the proof of Lemma 3.13. p -BSS-Tucker[ T ] ≤ p -BSS: Pick ε < ( np ) . Define the circuit C using the procedure described inLemma 4.6 and the homeomorphism φ of B n ( p − ) and P . Note that C is L -Lipschitz-continuousfor some L = O ( m ) . Thus, a solution of this instance of p -BSS is:1. a point x ∈ S n ( p − ) − such that C ( α x ) (cid:54) = α C ( x ) . In this case, let σ be the simplex such that φ − ( x ) ∈ (cid:107) σ (cid:107) , then there exists a vertex v ∈ V ( σ ) such that λ ( θ v ) (cid:54) = ( λ ( v ) + λ ( v )) .Observe that σ = value ( index ( φ − ( x ))) and that it has at most n ( p − ) + v . 37. a point x ∗ ∈ B n ( p − ) such that (cid:107)C ( x ∗ ) (cid:107) ∞ ≤ ε . In this case, φ − ( x ∗ ) must lie in a simplex witha fully labeled ( p − ) -dimensional face; this follows from the choice of ε and the proof ideasof Lemma 4.6. Observe that φ − ( C ( x ∗ )) is the convex combination of at most n ( p − ) + e i , j ’s. Hence, there must be a vector e i ∗ , j ∗ that appears in the convex combinationwith coefficient at least n ( p − )+ . If there is an i (cid:48) such that e i (cid:48) , j ∗ does not appear in the convexcombination, then φ − ( C ( x ∗ )) has at least one coordinate at least as large as n ( p − )+ . Hence, (cid:13)(cid:13) φ − ( C ( x ∗ )) (cid:13)(cid:13) ≥ (cid:13)(cid:13) φ − ( C ( x ∗ )) (cid:13)(cid:13) ∞ ≥ n ( p − )+ , which means that (cid:107)C ( x ∗ ) (cid:107) ≥ n ( p − )+ . Thisis a contradiction since it implies that (cid:107)C ( x ∗ ) (cid:107) ∞ ≥ (cid:107)C ( x ∗ ) (cid:107) √ np ≥ ( np ) > ε .The point φ − ( x ∗ ) lies in the simplex σ = value ( index ( φ − ( x ))) , which has at most n ( p − ) + ( p − ) -dimensional fully labeled face can be done efficiently. p -BSS ≤ p -BSS-Tucker[ T ]: Set m in T such that 1/2 m ≤ ε nL . The labeling λ is defined asin Lemma 4.7 using as function g : P → P the function given by φ − ◦ C ◦ φ , where φ is thehomeomorphism of B n ( p − ) and P ; note that all operations can be described with an arithmeticcircuit. A solution of this instance of p -BSS-T ucker [ T ] is:1. a vertex x ∈ ∂ T such that λ ( θ x ) (cid:54) = ( λ ( x ) + λ ( x )) . In this case, it must hold that C ( α ◦ φ ( x )) (cid:54) = α C ( φ ( x )) . Thus, φ ( x ) is a solution of p -BSS.2. a simplex σ ∗ ∈ T such that λ ( σ ∗ ) = Z p × { j } for some j ∈ [ n ] . In this case, following theproof of Lemma 4.7, there exists a vertex x in σ ∗ such that (cid:107) g ( x ) (cid:107) ∞ ≤ ε (or a violation of L -Lipschitz-continuity). Then, since φ as defined in Section 4.1 preserves the (cid:96) distancesbetween P and B n ( p − ) , (cid:107)C ( φ ( x )) (cid:107) ∞ ≤ (cid:107)C ( φ ( x )) (cid:107) ≤ ε . Thus, φ ( x ) is a solution of p -BSS. Remark p -BSS-T ucker ) . In order to use Kuhn’s triangulation we workon the domain C ∞ = { ( c , . . . , c p ) ∈ ([
0, 1 ] n ) p |∀ j ∈ [ n ] , ∃ i ∈ [ p ] : c ij = } instead of P . These arecoordinates with respect to the vectors e i , j . Note that C ∞ ∼ = P ∞ , where P ∞ = { ∑ i , j c ij e i , j | ( c , . . . , c p ) ∈ C ∞ } , and clearly P ∞ ∼ = P .We can triangulate C ∞ by using Kuhn’s triangulation (Definition 18) to triangulate each cube { ( c , . . . , c p ) ∈ C ∞ |∀ j ∈ [ n ] : c i j j = } for each ( i , . . . , i n ) ∈ [ p ] n . By the properties of Kuhn’striangulation, it immediately follows that this yields a triangulation of C ∞ . In particular, thetriangulations of two cubes “match” on their common subspace. Since we constructed thetriangulation separately on each cube, it follows that it refines the triangulation T ∗ . Furthermore,for any simplex σ lying on the boundary of C ∞ , it follows that θσ is also a simplex of thetriangulation. This is easy to see, because θ just changes the order of the coordinates, and theKuhn triangulation is invariant with respect to such transformations by definition. Thus, Kuhn’striangulation is indeed a nice triangulation. p -complete Having defined the computational problems corresponding to the BSS Theorem and to BSS-T ucker , we are ready to prove Theorem 4.1. The following theorem follows from Proposition 5.4,presented in Section 5.
Theorem 4.9.
For any prime p, p-
BSS-T ucker [ T ] is in PPA- p. p -BSS-T ucker is PPA- p -hard, through a reduction from p - polygon -T ucker . Theorem 4.10.
For any prime p ≥ , p- BSS-T ucker is PPA- p-hard, even for fixed dimension n ≥ . Note that for p = p -BSS-T ucker corresponds to the standard version of Tucker’s lemma,which is known to be PPA-hard for any n ≥ Proof.
Here we prove that p -BSS-T ucker is PPA- p -hard for n =
1. The hardness for any n ≥ n to n + n = p - polygon -T ucker to p -BSS-T ucker .Instead of the triangulation we used in the presentation of p - polygon -T ucker , we will use Kuhn’striangulation (Definition 18). It can be shown that the hardness of p - polygon -T ucker proved inProposition 3.15, also holds if we use Kuhn’s triangulation, since there is a simple homeomorphismbetween the two domains. We omit the details for this.Let λ be an instance of p - polygon -T ucker with Kuhn’s triangulation of size m . Thus, thedomain for this problem can be written as A = { ( c , . . . , c p ) ∈ ( U m ) p |∃ i ∈ [ p ] , ∀ j / ∈ { i , i + } : c j = } .We construct an instance of p -BSS-T ucker with n = C ∞ with Kuhn’striangulation of size m . Recall that the set of vertices can be written as C ∞ : = { ( c , . . . , c p ) ∈ U pm |∃ i ∈ Z p : c i = } . Let D : = ∪ i ∈ Z p D i , where D i : = { ( c , . . . , c p ) ∈ C ∞ |∀ j / ∈ { i , i + } : c j = } .We are going to embed the p - polygon -T ucker domain A into D in the most natural way. Sincewe are using Kuhn’s triangulation, the restriction of the triangulation of C ∞ to D correspondsto Kuhn’s triangulation on that domain, and thus to Kuhn’s triangulation of A . We define λ (cid:48) : C ∞ → Z p : λ (cid:48) ( c , . . . , c p ) = (cid:26) λ ( c , . . . , c p ) if ( c , . . . , c p ) ∈ DT p ( { j : c j = } ) otherwisewhere T p is the Z p -equivariant tie-breaking function defined in Definition 19. Note that λ (cid:48) iswell-defined, because if ( c , . . . , c p ) ∈ D , then it corresponds to a point in the p - polygon -T ucker domain. Furthermore, if ( c , . . . , c p ) / ∈ D , then |{ j : c j = }| ∈ [ p − ] , so is a valid input to T p .Since θ D = D , λ ( θ c ) = λ ( c ) + T p ( { j : ( θ c ) j = } ) = T p ( { j : c j = } ) +
1, it follows that λ (cid:48) satisfies the boundary conditions.Let c , c , . . . , c p be a ( p − ) -simplex of C ∞ that carries all the labels in Z p (with respect to λ (cid:48) ). We now show how this yields a solution to the p - polygon -T ucker instance. Without lossof generality, we can assume that c , c , . . . , c p are ordered in the order in which the simplex isdefined by Kuhn’s triangulation. Namely, for every i ∈ {
1, . . . , p − } , there exists j i such that c i + j i = c ij i + m and c i + j = c ij for all j (cid:54) = j i . Furthermore, the j i are all distinct. Note that if forsome i ∗ , c i ∗ / ∈ D , then c i / ∈ D for all i ≥ i ∗ . The following cases can occur:• the simplex does not intersect D : it follows that c (cid:54) = ∈ D , and thus there exists j suchthat c j >
0. But then c ij > i and the label j cannot be obtained by any vertex of thissimplex.• the intersection of the simplex with D is a face of dimension 2: then the three vertices of thisface have pairwise distinct labels, and thus yield a solution to p - polygon -T ucker .39 the intersection of the simplex with D is a face of dimension 1: then it must hold that c , c ∈ D and c , . . . , c p / ∈ D . We distinguish between the two sub-cases: – c j > c j = j (cid:54) = j . Then, since c / ∈ D , it follows that j / ∈ { j − j , j + } .By definition of the j i , we have that c j > c j >
0, which implies that c , . . . , c p can only have labels in Z p \ { j , j } . As a result, c and c must have labels j and j (inany order). This yields a solution to p - polygon -T ucker , because the labels are distinctand non-consecutive. – there exists j ∗ such that c j ∗ > c j ∗ + > c j = j / ∈ { j ∗ , j ∗ + } . Since c / ∈ D ,it must hold that j / ∈ { j ∗ , j ∗ + } . Thus we have c j >
0, and the vertices c , . . . , c p canonly obtain the labels Z p \ { j ∗ , j ∗ + j } . It follows that the simplex cannot possibly befully labeled.• the intersection of the simplex with D is a face of dimension 0: then c / ∈ D , and thus thereexist distinct j , j (cid:48) (and non-consecutive, but we don’t need this here) such that c j > c j (cid:48) >
0. But then, the vertices c , . . . , c p can only obtain the labels Z p \ { j , j (cid:48) } . It follows thatthe simplex cannot possibly be fully labeled.This completes the proof. Z p - star -T ucker Lemma: Statement and PPA- p -completeness In this section, we introduce a Z p -generalization of Tucker’s Lemma. We further define theassociated computational problem Z p - star -T ucker , and show that it is PPA- p -complete. In thenext section, we will use this problem to prove the membership of p -thief Necklace Splitting inPPA- p .In Z p - star -T ucker , the coordinates of the vertices lie on a star-like domain. This domain wasused by Meunier [2014] for the first fully combinatorial proof of necklace splitting with p thieves.For any prime p and any m ≥
1, we define R p , m = { } ∪ {∗ i j : i ∈ Z p , j ∈ [ m ] } , where [ m ] = {
1, 2, . . . , m } . For ease of notation, we also let ∗ i = i ∈ Z p . The symbols ∗ , . . . , ∗ p should be interpreted as p different “signs” that will generalize the use of “ + ” and “ − ” in Tucker’sLemma. The way to picture R p , m is as follows: the point 0 lies at the center and there are p segments of length m leaving from 0 in p different directions. In that sense, we also call R p , m a p -star. The boundary of the p -star is the set of points ∗ m , . . . , ∗ p m .The Z p -action θ is defined on R p , m in the natural way, i.e., θ ( ∗ i j ) = ∗ i + j (recall that i ∈ Z p ).In particular, θ ( ) =
0. For any d ≥ θ can be extended to R dp , m : = ( R p , m ) d by simply applying θ separately to each coordinate. Note that θ is a free action when restricted to the boundary of R dp , m (i.e., the points that have at least one coordinate of the form ∗ · m ). See Figure 6.There is a very natural metric on R p , m . dist ( ∗ i j , ∗ i j ) is defined to be | j − j | if i = i ,and j + j otherwise. We let dist ∞ ( · , · ) denote the generalization to R dp , m (where we take themaximum). Finally, we triangulate the domain R dp , m by using Kuhn’s triangulation on everysubcube (see Remark 5 below for details). We can now state a Tucker’s lemma for this domain. Theorem 5.1 ( Z p -star Tucker’s Lemma) . Let p be prime and m , t ≥ , d = t ( p − ) , and T be Kuhn’striangulation of R dp , m . Let λ : R dp , m → R p , t \ { } be any labeling that satisfies λ ( θ x ) = θλ ( x ) for all Notice that the set R p , t \ { } is isomorphic to the set [ p ] × [ t ] . ∈ ∂ R dp , m . Then there exists a ( p − ) -simplex x , . . . , x p of T and j ∈ [ t ] such that λ ( x i ) = ∗ i j for alli ∈ [ p ] . In particular, by the properties of Kuhn’s triangulation, it holds that for every solution x , . . . , x p , we have dist ∞ ( x i , x k ) ≤ i , k ∈ [ p ] .Note that Z -star Tucker’s Lemma corresponds to the standard version of Tucker’s Lemma.Since we are interested in the computational aspect, we also define the naturally correspondingTFNP problem. Z p - star -T ucker : I nput : m , t ≥ d = t ( p − ) and a Boolean circuit computing a labeling λ : R dp , m → R p , t \ { } that satisfies λ ( θ x ) = θλ ( x ) for all x ∈ ∂ R dp , m O utput : A ( p − ) -simplex x , . . . , x p ∈ R dp , m and j ∈ [ t ] such that λ ( x i ) = ∗ i j for all i ∈ [ p ] Note that the property “ λ ( θ x ) = θλ ( x ) for all x ∈ ∂ R dp , m ” can be enforced syntactically. Thus, Z p - star -T ucker is not a promise problem.The main result of the section is the following theorem. Theorem 5.2.
For all primes p, Z p - star -T ucker is PPA- p-complete.
The proof of the theorem will follow from Theorem 5.3 and Proposition 5.4 below. Thehardness is obtained by a reduction from p -BSS-T ucker , which is PPA- p -hard for all primes p , asshown in Theorem 4.10. Remark . The domain R dp , m can be triangulated in a standard way as follows. We start by subdivid-ing the domain into hypercubes {∗ i ( a − ) , ∗ i a } × · · · × {∗ i d ( a d − ) , ∗ i d a d } for a , . . . , a d ∈ [ m ] and i , . . . , i d ∈ Z p . Then, we can use Kuhn’s triangulation on each hypercube.Similarly to Remark 4, the triangulation T of R dp , m has the following nice properties:1. The restriction of T on any sub-orthant of R dp , m (i.e., a subspace of the form A × A × · · · × A d , where A (cid:96) = {∗ i (cid:96) j : 0 ≤ j ≤ m } or A (cid:96) = { } ) yields a triangulation of that sub-orthant.2. On the boundary of R dp , m , the triangulation T is symmetric with respect to θ : for any simplex σ of T that lies on the boundary of R dp , m , the simplices θσ , θ σ , . . . , θ p − σ are also simplicesof T (that also lie on the boundary).3. T is computationally efficient, in the sense that we can perform pivoting and indexingoperations in polynomial time. Z p - star -T ucker is in PPA- p First, we prove the membership of Z p - star -T ucker in PPA- p . We have the following theorem. Theorem 5.3.
For all primes p, Z p - star -T ucker lies in PPA- p. The result is proved by reducing the problem to I mbalance - mod - p . In particular, this alsoprovides a combinatorial proof of Z p -star Tucker’s Lemma (Theorem 5.1). This proof can be seenas a generalization of the combinatorial proof of Tucker’s Lemma given by Freund and Todd[1981]. We provide an overview of the proof below; the full proof can be found in Appendix C.41igure 6: A view of the domain R dp , m for p = d =
2. Note that this corresponds to R m × R m .The three black points are in correspondence under θ . The three thick lines at the center of thepicture correspond to the place where the three pieces are “glued” together. As noted earlier, Z - star -T ucker corresponds to the standard version of Tucker’s Lemma andthe domain is equivalent to {− m , − ( m − ) . . . , 0, . . . , m } d . The computational problem is knownto lie in PPA (recall that PPA = PPA-2) by using an argument given by Freund and Todd [1981].More precisely, Freund and Todd gave a constructive proof of Tucker’s Lemma and as noted by[Papadimitriou, 1994, Aisenberg et al., 2020], this yields a reduction to PPA. The constructive proofrelies on a path-following argument on a graph where the nodes are simplices of the triangulation.We start by giving some details about their argument, since our proof is a generalization of theirconstruction.The nodes of the graph G consist of all simplices of the triangulation that satisfy someproperties that depend on the labels of the simplex (and the coordinate subspace orthant in whichthe simplex lies). Following the presentation of the proof given by Matoušek [2008], we call these“happy simplices”. Undirected edges are added between happy simplices based on some simplerules (e.g., if they share a facet and that facet has some desired labels, etc). Given the definition ofthe edges, it is easy to show that the happy simplex 0 d has degree 1 and any other node of degree1 is either a solution, or is a happy simplex lying on the boundary of the domain. In the lattercase, because of the boundary conditions, this means that there must be another such simplexthat lies on the antipodally opposite side of the domain and also has degree 1. Thus, mergingthese two nodes into a single one yields a vertex of degree 2, eliminating these “fake” solutions.Our first contribution is to note that the edges of the graph can be directed in a consistent wayin Freund and Todd’s construction. Namely, any non-merged degree-2 vertex has one incomingand one outgoing edge, and any merged vertex has either two incoming edges, or two outgoing42dges. This yields a reduction to I mbalance - mod -2. Note that if all degree 2 vertices were alwaysperfectly balanced, then we would obtain a reduction to E nd - of -L ine , which is impossible, unlessPPAD = PPA.When we move to the case p >
2, the notions used to define the graph can be generalizedin a natural way, despite the unusual domain R dp , m . While the ability to direct edges was notactually needed for p =
2, it now becomes absolutely necessary. Indeed, for any degree-1 happysimplex on the boundary, there are now p − θ ). Merging theseinto a single node yields degree p . We show that directing the edges yields a merged node that isbalanced modulo p (namely, all p edges are incoming, or they are all outgoing).However, another difficulty arises for p >
2. Recall that a path can visit simplices of variousdimensions. The vertices where the dimension changes are special vertices, that we call super-happy simplices. These super-happy simplices have one edge with a same or lower-dimensionalhappy simplex, and k edges with k different higher-dimensional happy-simplices, where k ∈ [ p − ] .Directing the edges as before, yields that the k edges are directed the same way, and in the oppositedirection to the single edge. By changing the way the direction of edges is defined, it is possibleto salvage the situation for p =
3. However, this fails for any p ≥ weights to all edges. The weight of an edge only depends onthe nature of the coordinate subspace orthants in which it lies, in particular the dimension. Withthese weights, we show that any vertex that is not a solution is now balanced modulo p (exceptthe trivial solution 0 d ). Namely, the non-solution vertices of the graph are:• the trivial solution 0 d : all its edges are outgoing and it has degree ( − ) t mod p • the merged simplices on the boundary: p edges, all incoming/outgoing, all the same weight• happy, but not super-happy simplices: once incoming edge, one outgoing, both same weight• super-happy simplices: one incoming edge with weight w and k ∈ [ p − ] outgoing edges,each with weight w / k (or opposite direction for all edges)Thus, apart from the trivial solution and any actual solutions, all vertices are balanced modulo p . Remark . Even though this proof is a natural generalization of theargument by Freund and Todd [1981], it is not a path-following argument for p ≥
3. Indeed, inthe case where p ≥
3, it is not clear how we could explore this graph by following a path that isguaranteed to end at a solution. In fact, we provide strong evidence that it is not possible to prove Z p - star -T ucker by a path-following argument. Since Z p - star -T ucker is PPA- p -hard (see nextSection), and since a path-following proof of Z p - star -T ucker would presumably show that theproblem lies in PPA, this would imply that PPA- p ⊆ PPA, for prime p ≥
3. However, this is notexpected to hold [Johnson, 2011, Göös et al., 2020, Hollender, 2019].Similarly, if one can show that p -N ecklace -S plitting is PPA- p -hard for some prime p ≥ p thieves cannotbe proved by a path-following argument. Z p - star -T ucker is PPA- p -hard In this section we prove that Z p - star -T ucker is PPA- p -hard, by reducing from p -BSS-T ucker ,which is PPA- p -hard by Theorem 4.10. 43 roposition 5.4. For any prime p, Z p - star -T ucker is PPA- p-hard, even for fixed dimension t ≥ .Proof. We show this by reducing from p -BSS-T ucker with n = t . Let λ be an instance of p -BSS-T ucker for some prime p and some n ≥
1, where we use Kuhn’s triangulation.In this case the domain of p -BSS-T ucker corresponds to C ∞ = { ( c , . . . , c p ) ∈ U npm |∀ j ∈ [ n ] , ∃ i ∈ [ p ] : c ij = } .On the other hand, the domain of Z p - star -T ucker can be described as A = { ( a , . . . , a p ) ∈ U np ( p − ) m |∀ j , k ∈ [ n ] × [ p − ] , ∃ i ∗ ∈ [ p ] : a ij , k = i ∈ [ p ] \ { i ∗ }} .Note that θ ( c , . . . , c p ) = ( c p , c , . . . , c p − ) and θ ( a , . . . , a p ) = ( a p , a , . . . , a p − ) .Define Ψ : A → C ∞ as Ψ ( a , . . . , a p ) = ( ψ ( a ) , . . . , ψ ( a p )) , where ψ j ( a i ) = max { a ij , k : k ∈ [ p − ] } for all i ∈ [ p ] , j ∈ [ n ] . Note that Ψ is well-defined, namely if a ∈ A , then Ψ ( a ) ∈ C ∞ .Indeed, for any j ∈ [ n ] , it holds that |{ i ∈ [ p ] | ψ j ( a i ) > }| = |{ i ∈ [ p ] |∃ k ∈ [ p − ] : a ij , k > }| ≤|{ ( i , k ) ∈ [ p ] × [ p − ] | a ij , k > }| ≤ p −
1, since for every k ∈ [ p − ] there exists at most one i ∈ [ p ] such that a ij , k > j ). Furthermore, it is easy to see that Ψ ( θ a ) = θ Ψ ( a ) byconstruction.Now define the labeling λ (cid:48) : A → Z p × [ n ] by λ (cid:48) ( a ) : = λ ( Ψ ( a )) . Since Ψ ( A ) ⊆ C ∞ , λ (cid:48) iswell-defined. Furthermore, for any a ∈ ∂ A , it holds that Ψ ( a ) ∈ ∂ C ∞ . Thus, we get that forany a ∈ ∂ A , λ (cid:48) ( θ a ) = λ ( Ψ ( θ a )) = λ ( θ Ψ ( a )) = θλ ( Ψ ( a )) = θλ (cid:48) ( a ) , i.e., λ (cid:48) satisfies the boundaryconditions.If σ = { z , . . . , z p } is a ( p − ) -simplex of A such that λ (cid:48) ( σ ) = Z p × { j } for some j ∈ [ n ] , thenthe set of vertices Ψ ( σ ) = { Ψ ( z ) , . . . , Ψ ( z p ) } satisfies λ ( Ψ ( σ )) = Z p × { j } . Thus, the proof iscompleted by the following claim: Claim . If σ = { z , . . . , z p } is a ( p − ) -simplex in Kuhn’s triangulation of A , then Ψ ( σ ) = { Ψ ( z ) , . . . , Ψ ( z p ) } is a simplex in Kuhn’s triangulation of C ∞ . Proof.
Since we use Kuhn’s triangulation, without loss of generality, we can assume that z , . . . , z p are ordered such that z ≤ z ≤ · · · ≤ z p (component-wise) and (cid:107) z − z p (cid:107) ∞ ≤ m . Byconstruction of Ψ it holds that Ψ ( a ) ≤ Ψ ( a (cid:48) ) whenever a ≤ a (cid:48) . Thus, Ψ ( z ) ≤ · · · ≤ Ψ ( z p ) . Inorder to show that Ψ ( σ ) is a simplex in Kuhn’s triangulation of C ∞ , it remains to prove that (cid:107) Ψ ( z ) − Ψ ( z p ) (cid:107) ∞ ≤ m . To see this, note that if (cid:107) a − a (cid:48) (cid:107) ∞ ≤ m , then for all i ∈ [ p ] and j ∈ [ n ] ,we get that | ψ j ( a i ) − ψ j ( a (cid:48) i ) | = | max { a ij , k : k ∈ [ p − ] } − max { a (cid:48) ij , k : k ∈ [ p − ] }| ≤ m .This concludes the proof of Proposition 5.4. p Thieves lies in PPA- p In this section, we prove our main result regarding the Necklace Splitting Theorem; we prove thatthe associated computational problem p -N ecklace -S plitting lies in PPA- p for any prime p . Theorem 6.1.
For every prime p, p -N ecklace -S plitting is in PPA- p. As a corollary, we also obtain that:• p r -N ecklace -S plitting is in PPA- p r for any prime p and r ≥ k -N ecklace -S plitting lies in PPA- k under Turing reductions for any k ≥ ε -C onsensus -1/ p -D ivision problem, which is the computational analogue of theConsensus-1/ p -Division problem of Simmons and Su [2003]. We will show the inclusion of ε -C onsensus -1/ p -D ivision in PPA- p via a reduction to Z p - star -T ucker , which also implies thePPA- p membership for p -N ecklace -S plitting . We prove the following main statement: Theorem 6.2.
For any prime p, C onsensus -1/ p -D ivision reduces to Z p - star -T ucker . The proof is presented in Section 6.1, where we prove an even stronger version of this theorem.Indeed, we show that this result holds for any probability measures (not only step functions), aslong as they are efficiently computable and sufficiently continuous (in some precise sense).We start with the complete definitions of the computational problems corresponding to k -thiefNecklace Splitting and Consensus-1/ k -Division. k -N ecklace -S plitting [Papadimitriou, 1994, Filos-Ratsikas and Goldberg, 2019]I nput : An open necklace with n beads, each of which has one of t colors.There are exactly a i k beads of color i =
1, . . . , t , where a i ∈ N .O utput : A partitioning of the necklace into k (not necessarily connected) pieces such that eachpiece contains exactly a i beads of color i , using at most ( k − ) t cuts.As proved by Alon [1987], this problem always has a solution. Furthermore, for any proposedsplitting of the necklace it is easy to check if it is a solution. Thus, the problem k -N ecklace -S plitting lies in TFNP.Alon [1987] proved the necklace-splitting theorem by showing existence for a more generalcontinuous version and then “rounding” a solution of the continuous problem to obtain a splittingof the necklace. When investigating the complexity of k -N ecklace -S plitting , it is also convenientto consider the more general continuous version. Even though the continuous theorem is termedas a “generalized Hobby-Rice theorem” by Alon [1987], we instead use the term Consensus- k-Division proposed by Simmons and Su [2003]. ε -C onsensus -1/ k -D ivision [Filos-Ratsikas et al., 2018, Filos-Ratsikas and Goldberg, 2018]I nput : ε > µ , . . . , µ t on [
0, 1 ] .The probability measures are given by their density functions on [
0, 1 ] , which are step functions(explicitly given in the input).O utput : A partitioning of the unit interval into k (not necessarily connected) pieces A , . . . , A k using at most ( k − ) t cuts, such that | µ j ( A i ) − µ j ( A (cid:96) ) | ≤ ε for all i , j , (cid:96) .The fact that this problem also lies in TFNP immediately follows from showing that itlies in PPA- k under Turing reductions (Theorem 6.4). Note that we can equivalently ask for µ j ( A i ) = k ± ε for all i , j . Indeed, the computational problems are equivalent.Furthermore, using the same technique as Etessami and Yannakakis [2010, Theorem 5.2], onecan show that if ε is sufficiently small (with respect to the representation size of the step functions),45hen one can efficiently compute an exact solution from an ε -approximate solution. It followsthat ε -C onsensus -1/ k -D ivision is equivalent to exact C onsensus -1/ k -D ivision . In particular,the problem always has an exact solution that is rational. Thus, we will sometimes refer to thisproblem just as C onsensus -1/ k -D ivision .Alon’s rounding procedure yields a reduction from k -N ecklace -S plitting to exact C onsensus -1/ k -D ivision . Filos-Ratsikas and Goldberg [2018] extended this result by showing that k -N ecklace -S plitting reduces to ε -C onsensus -1/ k -D ivision , even when ε is not small enough toensure that we can get an exact solution. Proposition 6.3 (Alon [1987], Filos-Ratsikas and Goldberg [2018]) . For any k ≥ , k -N ecklace -S plitting reduces to ε -C onsensus -1/ k -D ivision . Before we proceed with the proof of Theorem 6.2, we present the consequences of Theorems 5.3and 6.2, in terms of computational complexity and mathematical existence.
Consequences: Computational ComplexityTheorem 6.4.
For any k ≥ , k -N ecklace -S plitting and C onsensus -1/ k -D ivision lie in the Turingclosure of PPA- k. In particular, if k = p r where p is prime and r ≥ , then the problems lie in PPA- p. The Turing closure of PPA- k is the class of all TFNP problems that Turing-reduce to a PPA- k -complete problem (e.g., I mbalance - mod - k ). Note that when k is not a prime power, PPA- k is notbelieved to be closed under Turing reductions [Göös et al., 2020, Hollender, 2019]. Proof.
Theorem 5.3 and Theorem 6.2 immediately imply that for any prime p , C onsensus -1/ p -D ivision lies in PPA- p . As noted by Alon [1987, Proposition 3.2], for any k , (cid:96) ≥
2, a C onsensus -1/ ( k (cid:96) ) -D ivision can be obtained by first finding a C onsensus -1/ k -D ivision – which divides theinterval into k (not necessarily connected) pieces – and then finding a C onsensus -1/ (cid:96) -D ivision of each of the k pieces. Note that we obtain (at most) the desired number of cuts. Thus, we cansolve an instance of C onsensus -1/ ( k (cid:96) ) -D ivision by first solving an instance of C onsensus -1/ k -D ivision , and then k instances of C onsensus -1/ (cid:96) -D ivision .In particular, for any prime p and any r ≥
1, C onsensus -1/ p r -D ivision can be solved bysolving 1 + p + p + · · · + p r − instances of C onsensus -1/ p -D ivision . Thus, we obtain a Turingreduction from C onsensus -1/ p r -D ivision to C onsensus -1/ p -D ivision . Since C onsensus -1/ p -D ivision lies in PPA- p and PPA- p is closed under Turing reductions [Göös et al., 2020, Hollender,2019], it follows that C onsensus -1/ p r -D ivision lies in PPA- p .Now consider any k = ∏ mi = p r i i , where m ≥ r i ≥ p i are distinct primes. Then,C onsensus -1/ k -D ivision can be solved by a query to C onsensus -1/ p r -D ivision , then p r queriesto C onsensus -1/ p r -D ivision (which can be turned into a single query to PPA- p ), then p r p r queries to C onsensus -1/ p r -D ivision (which can also be turned into a single query to PPA- p ),etc. Thus, C onsensus -1/ k -D ivision can be solved by a query to PPA- p , then a query to PPA- p ,then one to PPA- p , . . . , and finally a query to PPA- p m . Since PPA- p i ⊆ PPA- k for i =
1, . . . , m (Proposition 2.2), it follows that there is a Turing reduction from C onsensus -1/ k -D ivision to aPPA- k -complete problem (e.g., I mbalance - mod - k ).Since k -N ecklace -S plitting reduces to C onsensus -1/ k -D ivision (Proposition 6.3), the resultsalso hold for k -N ecklace -S plitting . 46 onsequences: Mathematical Existence Theorems 5.3 and 6.2 yield a reduction from C onsensus -1/ p -D ivision to I mbalance - mod - p .Since every instance of I mbalance - mod - p has a solution (and the proof of this is trivial), weobtain a proof that C onsensus -1/ p -D ivision always has a solution. Thus, this proves that ifthe probability measures are step functions (described by rational numbers), there always existsa consensus-1/ p -division. While we have given the proof in terms of a reduction (since this isrequired for our complexity results), it can also be written as a mathematical proof of existence(without any computational considerations).Once existence of a consensus-1/ p -division for step functions has been proved, a constructiveargument by Alon [1987, Section 2] also gives existence for p -necklace-splitting. Putting everythingtogether, the proof of p -necklace-splitting thus obtained is a fully combinatorial proof that doesnot use any advanced machinery and is easier to follow than existing proofs. Indeed, as wementioned in the introduction, the original proof by Alon [1987] used the BSS theorem of Bárányet al. [1981] as a black box. The only other fully combinatorial proof by Meunier [2014], whilequite elegant, is significantly more involved.Going back to consensus-1/ p -division, the proof we obtain (which uses Z p -star Tucker’slemma) actually works for any probability measures, not only step functions. Moreover, similarlyto Simmons and Su [2003], our proof does not make use of the fact that the measures are additiveand non-negative. Thus, we obtain a stronger version of the consensus-1/ p -division theoremgiven by Alon [1987, Theorem 1.2] (which he calls a generalization of the Hobby-Rice theorem).Let B ([
0, 1 ]) denote the Borel σ -algebra on the unit interval and let λ denote the Lebesgue measureon the unit interval. Finally, let (cid:52) denote the symmetric difference, i.e., A (cid:52) B = ( A \ B ) ∪ ( B \ A ) . Theorem 6.5.
Let p be any prime and t ≥ . Let v , . . . , v t : B ([
0, 1 ]) → R be such that for all ≤ j ≤ tv j satisfies the following continuity condition: for all ε > there exists δ > such that | v j ( A ) − v j ( B ) | ≤ ε for all A , B ∈ B ([
0, 1 ]) that satisfy λ ( A (cid:52) B ) ≤ δ . Then, there exists a consensus- p-division. Namely, it is possible to partition the unit interval into p (notnecessarily connected) pieces A , . . . , A p using at most ( p − ) t cuts, such that v j ( A i ) = v j ( A (cid:96) ) for all ≤ i , (cid:96) ≤ p, ≤ j ≤ t. Before we move on to the proof, let us briefly explain why we only obtain the result for prime p . In the usual setting where the valuations are probability measures, it is enough to prove thestatement for primes. Indeed, using the standard argument by Alon [1987, Proposition 3.2], ifa consensus-1/ k -division and a consensus-1/ (cid:96) -division always exist, then a consensus-1/ ( k (cid:96) ) -division exists. But Alon’s argument makes use of the additivity property of the measures. Indeed,consider the non-additive setting and say that we are trying to show that a consensus-1/4-exists.We know that a consensus-1/2-division exists and this yields a partition of [
0, 1 ] into A and A .We have that v j ( A ) = v j ( A ) for all j . Following Alon’s argument, find a consensus-1/2-divisionof A and of A . This yields A ∪ A = A and A ∪ A = A such that v j ( A ) = v j ( A ) and v j ( A ) = v j ( A ) for all j . However, we might not have v j ( A ) = v j ( A ) , since we no longerhave v j ( A ) + v j ( A ) = v j ( A ) + v j ( A ) .If we assume that the valuations are additive (even just finite additivity), but still allow themto take negative values, then Alon’s argument works as before and thus the result again holds forany k ≥
2. 47 roof.
Using Z p -star Tucker’s lemma (Theorem 5.1) it follows that for any ε >
0, there exists an ε -approximate consensus-1/ p -division, i.e., we can partition the interval into p pieces A , . . . , A p (using at most ( p − ) t cuts) such that | v j ( A i ) − v j ( A (cid:96) ) | ≤ ε for all 1 ≤ i , (cid:96) ≤ p , 1 ≤ j ≤ t . Indeed,it suffices to follow the same steps as in the proof of Theorem 6.6.Let R p denote the continuous version of R p , m . R p consists of p copies of the segment [
0, 1 ] (namely ∗ [
0, 1 ] , . . . , ∗ p [
0, 1 ] ) that share the same origin (i.e., ∗ = · · · = ∗ p R dp corresponds to a way topartition [
0, 1 ] into p (not necessarily connected) pieces using at most ( p − ) t cuts. Since ( R dp , dist ∞ ) is a compact metric space, every sequence must have a subsequence that converges.Thus, a sequence ( x n ) n , where x n ∈ R dp is a 1/2 n -approximate consensus-1/ p -division, must havea subsequence that converges to some x ∈ R dp . For any i , j the function f : R dp → R , y (cid:55)→ v j ( A i ( y )) is continuous. It follows that x must correspond to an exact consensus-1/ p -division. C onsensus -1/ p -D ivision to Z p - star -T ucker In order to make our PPA- p -membership result as strong as possible, we define a computationalproblem that is much more general than ε -C onsensus -1/ p -D ivision . Namely, we allow anycomputationally reasonable probability measures that are also sufficiently continuous.The probability measures are given by their cumulative functions. Let F be a class ofcumulative distribution functions on [
0, 1 ] . Thus, for any f ∈ F and any a ∈ [
0, 1 ] , f ( a ) is theprobability of the interval [ a ] according to f . For any f ∈ F , let size ( f ) denote the size of therepresentation of f . E.g., if f is represented as a circuit, then size ( f ) is the size of the circuit. Forany rational number x , size ( x ) denotes the representation length of x , i.e., the length of the binaryrepresentation of the denominator and numerator of x . We will require two properties from F :• F is polynomially computable: there exists a polynomial q such that for all f ∈ F and allrational x ∈ [
0, 1 ] , f ( x ) can be computed in time q ( size ( f ) + size ( x )) .• F is polynomially continuous: there exists a polynomial q such that for all f ∈ F andall rational (cid:98) ε >
0, there exists rational (cid:98) δ > ( (cid:98) δ ) ≤ q ( size ( f ) + size ( (cid:98) ε )) such that | x − y | ≤ (cid:98) δ = ⇒ | f ( x ) − f ( y ) | ≤ (cid:98) ε for all x , y ∈ [
0, 1 ] .These properties are quite natural and they were used by Etessami and Yannakakis [2010] in thecontext of fixed point problems. In particular, they hold when F is the class of all cumulativedistribution functions given by step function densities (represented explicitly). But they also holdfor much more general families. Definition 10.
Let k ≥ F be a polynomially computable and polynomially continuousclass of cumulative distribution functions on [
0, 1 ] . The problem ε -C onsensus -1/ k -D ivision [ F ] isdefined exactly as ε -C onsensus -1/ k -D ivision , except that the probability measures are given bycumulative distribution functions in F .Notice that ε -C onsensus -1/ k -D ivision corresponds to the special case where F is the classof all cumulative distribution functions given by step function densities (represented explicitly).Thus, the following is a stronger version of Theorem 6.2. Theorem 6.6.
Let p be prime and F be a polynomially computable and polynomially continuous class ofcumulative distribution functions on [
0, 1 ] . Then ε -C onsensus -1/ p -D ivision [ F ] reduces to Z p - star -T ucker . roof. Let ε > µ , . . . , µ t be probability measures on [
0, 1 ] given by functions in F . Weconsider the domain D = R dp , m , where d = t ( p − ) and m ≥ D represents a way to partition [
0, 1 ] into p (not necessarily connected) pieces using at most t ( p − ) cuts. This is a slight modification of the domain that was used by Meunier [2014] to encode asplitting of a necklace. Intuitively it can be explained as follows. Let x = ( ∗ i j , . . . , ∗ i d j d ) ∈ D . Weinterpret each element i ∈ Z p as a different color. Then:1. Paint the whole interval [
0, 1 ] with the color 1.2. For (cid:96) =
1, 2, . . . , d : paint [ j (cid:96) / m ] with the color i (cid:96) Note that applying a fresh coat of paint on a previously painted part of the interval covers upthe old paint. The way the interval [
0, 1 ] is colored at the end of this procedure gives us thepartition encoded by x ∈ D . An important advantage of this encoding is that it is sufficientlycontinuous in a certain sense. Indeed, small changes in the coordinates of x have a small effect onthe corresponding partition. Other simpler encoding schemes do not have this property.Formally, this encoding can be described as follows. Add a “fake” 0th coordinate ∗ i j = ∗ m . Place cuts at all positions j / m , j / m , . . . , j d / m . This subdivides the interval [
0, 1 ] into atmost d + = t ( p − ) + [ a , b ] to i (cid:98) (cid:96) ∈ Z p , where (cid:98) (cid:96) = max { ≤ (cid:96) ≤ d : j (cid:96) / m ≥ b } .This encoding also behaves nicely with respect to θ . For any x ∈ ∂ D , θ x encodes the samepartition as x , except that i has been replaced by i +
1, for all i ∈ Z p . This is easy to see since forany x ∈ ∂ D , there exists (cid:96) ≥ j (cid:96) = m and thus the “fake” coordinate ∗ i j does not playany role.We are now ready to define the labeling λ : D → R p , t \ { } . This labeling is a naturalgeneralization of the one used by Simmons and Su [2003]. Given x ∈ D , construct the partition itencodes, namely A ( x ) , . . . , A p ( x ) . Then, for all i ∈ Z p and j ∈ [ t ] , let µ j , i ( x ) = µ j ( A i ( x )) , i.e., thetotal measure of type j that is allocated to i . Finally, set λ ( x ) = ∗ i j , where i , j are determined asfollows:1. Pick j ∈ [ t ] that maximizes max i , i | µ j , i ( x ) − µ j , i ( x ) | . Break ties by picking the minimumsuch j .2. Then, pick i ∈ Z p that maximizes µ j , i ( x ) . If there are multiple i ’s that maximize this, breakties by picking the one such that min A i ( x ) is minimal (i.e., such that A i ( x ) contains thepoint closest to the left end of the unit interval).By using the observation above about the behavior of θ on ∂ D , it is easy to see that λ ( θ x ) = θλ ( x ) for all x ∈ ∂ D . Thus, λ is a valid instance of Z p - star -T ucker and we obtain a solution x , . . . , x p ∈ D and (cid:98) ∈ [ t ] , such that dist ∞ ( x i , x k ) ≤ λ ( x i ) = ∗ i (cid:98) for all i , k ∈ [ p ] . It remainsto show that by picking m large enough, we obtain a solution to ε -C onsensus -1/ p -D ivision [ F ].Let ε (cid:48) = ε t ( p − ) . Since F is polynomially continuous, we can pick m large enough so thatthe value µ j ([ a ]) changes by at most ε (cid:48) , if a moves by 1/ m . Note that m has representationlength polynomial in the size of the instance. If a single coordinate of x changes by 1, µ j , i ( x ) changes by at most ε (cid:48) for all i , j . Since there are d = t ( p − ) coordinates, it follows that | µ j , i ( x k ) − µ j , i ( x (cid:96) ) | ≤ ε (cid:48) t ( p − ) = ε /2 for all i , j and for all k , (cid:96) ∈ [ p ] .49et x : = x . By construction of the labeling, we obtain that for all j , i , (cid:96) | µ j , i ( x ) − µ j , (cid:96) ( x ) | ≤ max i , i | µ (cid:98) , i ( x ) − µ (cid:98) , i ( x ) | ≤ ε The first inequality holds because λ ( x ) = ∗ (cid:98) . The second inequality holds because if weinstead had µ (cid:98) , i ( x ) > µ (cid:98) , i ( x ) + ε for some i , i , then it would follow that µ (cid:98) , i ( x i ) > µ (cid:98) , i ( x i ) ,contradicting λ ( x i ) = ∗ i (cid:98) . Thus, x corresponds to an ε -approximate solution. Our topological characterization of PPA- p can possibly enable us to obtain similar membership orhardness results for other interesting problems. For example, are the problems whose totality isestablished via the BSS Theorem, like the Chromatic Number of Kneser Hypergraphs studiedin [Alon et al., 1986] in PPA- p ? Are they PPA- p -complete? We believe that due to its simplicity,our p - polygon -T ucker problem can be a very useful tool for obtaining hardness results for theseproblems. What about other problems that generalize problems that are known to be in PPA orare even PPA-complete? For example, Filos-Ratsikas and Goldberg [2019] showed that the discreteHam-Sandwich problem is also PPA-complete. Is there a generalization of the problem that couldbe complete for PPA- p ? A computational version of the Center Transversal Theorem [Dol’nikov,1992, Zivaljevi´c and Vre´cica, 1990] might be a good candidate. Another interesting open problemis to investigate the connection of the general statement of Dold’s Theorem [Dold, 1983] fromalgebraic topology with the subclasses of TFNP. Finally, although our paper takes a definitivestep in the direction of resolving the complexity of p -thief Necklace Splitting and Consensus-1/ p -Division, proving a PPA- p -hardness result remains a challenging open problem. In veryrecent work [Filos-Ratsikas et al., 2020], we have made a first step in that direction, by providing asignificantly simpler proof (and strengthening) of the PPA-2-hardness result of Filos-Ratsikas andGoldberg [2019], as well as the first hardness result for Consensus-1/3-Division, showing that it ishard for the class PPAD. Showing the PPAD-hardness of the Consensus-1/ p -Division problem for p > p -hardness. Acknowledgments
Alexandros Hollender is supported by an EPSRC doctoral studentship (Reference 1892947). Kate-rina Sotiraki is supported in part by NSF/BSF grant
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Topological Definitions
In this section, we include all the necessary notation and topological definitions that are usedthroughout the paper. For a more detailed exposition on simplicial complexes, we refer theinterested reader to [Matoušek, 2008].
Notation:
Let B n = { x ∈ R n : (cid:107) x (cid:107) ≤ } denote the n -dimensional unit ball and S n − = ∂ B n bethe corresponding unit sphere. Definition 11 (H omeomorphism ) . A homeomorphism of topological spaces ( X , O ) and ( X , O ) is a bijection φ : X → X such that for every U ⊆ X , φ ( U ) ∈ O if and only if U ∈ O . In otherwords, a bijection φ : X → X is a homeomorphism if and only if both φ and φ − are continuous.If there is a homeomorphism φ : X → X , we write X ∼ = Y .We say that a function f has order p if f p = f , where the notation f i denotes to the compositionof f by itself i times. Definition 12 (F ree A ction ) . Let f : X → Y be a function of order p and let P be a set. We saythat f acts freely on P if for all x ∈ X and all i ∈ {
1, . . . , p − } , f i ( x ) (cid:54) = x . Definition 13 (A ffine I ndependence ) . We call the points v , v , . . . , v k affine dependent if thereexist numbers a , a , . . . , a k ∈ R not all 0 such that k ∑ i = a i v i = and k ∑ i = a i =
0. Otherwise, v , . . . , v k are called affine independent .Geometrically, some examples of simplices are points, lines and triangles. Formally, thedefinition requires the notion of affine independence. Definition 14 (S implex ) . A simplex σ is the convex hull of a finite set A of affine independentvectors in R n . The points in A are called the vertices of σ and denoted by V ( σ ) . The dimensionof σ is equal to | A | −
1. Namely, a k dimensional simplex, called k -simplex for short, has k + σ is called a face of σ . A properface of σ is called facet .From the above definitions, it holds that every face is itself a simplex. For simplicity, we denotea simplex as the set of its vertices. A.1 Simplicial Complexes, Value & Index Functions and Triangulations
Very central to our paper is the notion of geometric simplicial complexes , which are used to describesubspaces of R d . These subspaces consist of simple building blocks, such as points, line segments,triangles, tetrahedra, that are pasted together. Definition 15 (S implicial C omplex ) . A simplicial complex K is a non-empty set of simplices thatsatisfies the following properties:• Each face of a simplex σ ∈ K is also a simplex in K .• The intersection σ ∩ σ of any two simplices σ , σ ∈ K is a face of both σ and σ .55igure 7: A simplicial complex T that defines a triangulation of the triangle A − B − C .The union of the simplices in K is called the polyhedron of K and is denoted by (cid:107) K (cid:107) . Thedimension of K is dim ( K ) : = max σ ∈ K { dim ( σ ) } and the vertex set of K , denoted by V ( K ) , is theunion of the vertex sets of all its simplices.We denote by Σ σ for a simplex σ the simplicial complex that contains all simplices τ such that τ ⊆ σ .According to the above definition, zero-dimensional simplicial complexes correspond to pointsand one-dimensional simplicial complexes to sets of non-intersecting line segments as shown inFigure 7.The notion of triangulation relates simplicial complexes with topological spaces. Definition 16 (T riangulation ) . A simplicial complex K is a triangulation of a topological space X if (cid:107) K (cid:107) ∼ = X .For instance, the boundary of the n -simplex σ n , namely a simplicial complex containing allproper faces of σ n , is a triangulation of the sphere S n − .Triangulation is a very powerful tool in studying the computational complexity of topologicalproblems, because they allow us to partition a simplicial complex into smaller simplices that areconnected in useful ways. We will mainly use the Kuhn triangulation , which is described in moredetail below. We refer the interested reader to [Matoušek, 2008] and [Munkres, 1984] for furtherinformation.We define two functions of a triangulation, index and value , that are essential for the definitionof our computational problems and our reductions.
Definition 17 (V alue & I ndex F unctions ) . Let K be a simplicial complex consisting of k simplices,including the non-full dimensional ones and let M > k . We define the value function value : [ M ] → ¯ K , where ¯ K = K ∪ { ∅ } , to be an efficiently computable function such that1. value is bijective on K ,2. if value ( x ) = { v , . . . , v (cid:96) } , then x is called the index of the simplex σ ∈ K with vertices { v , . . . , v (cid:96) } , and 56. if value ( x ) = ∅ then x does not correspond to a valid index of any non-empty simplex in K ,We also define the index function index : R n → [ M ] to be an efficiently computable functionsuch that if x ∈ (cid:107) K (cid:107) then x ∈ (cid:107) value ( index ( x )) (cid:107) .Intuitively, value provides a way to enumerate over the simplices and index on input a point x returns the simplex that contains x . Definition 18 (K uhn ’ s T riangulation [Kuhn, 1960]) . Kuhn’s triangulation is a standard way totriangulate a domain that is a cube. For any n ∈ N , the cube [
0, 1 ] n is triangulated with granularity m ∈ N as follows:1. the set of vertices of the triangulation is U nm , where U m = {
0, 1/ m , 2/ m , . . . , m / m }
2. every x ∈ ( U m \ { } ) n is the base of the cube containing all vertices { y : y i ∈ { x i , x i + m }}
3. every such cube is subdivided into n ! n -dimensional simplices as follows: for every permu-tation π of [ n ] , σ = { y , y , . . . , y n } is a simplex, where y = x and y i = y i − + m e π ( i ) for all i ∈ [ n ] ( e i is the i th unit vector)It is easy to see that Kuhn’s triangulation has the following properties:• For any simplex σ = { z , . . . , z k } it holds that (cid:107) z i − z j (cid:107) ∞ ≤ m for all i , j , and there exists apermutation π of [ k ] such that z π ( ) ≤ · · · ≤ z π ( k ) (component-wise).• The restriction of Kuhn’s triangulation of [
0, 1 ] n on some subspace A × A × · · · × A n of [
0, 1 ] n , where for each i ∈ [ n ] , A i ∈ {{ } , [
0, 1 ] } , coincides with Kuhn’s triangulation of thatsubspace.• Every n -dimensional simplex can be indexed by its smallest vertex (component-wise), whichis also the base of the cube containing the simplex, and by the permutation that yields thissimplex within this cube. Given some index, it is easy to check whether this is a valid index,and if so, output the vertices of the simplex. Thus, the index function can be computedefficiently.• Given a point x ∈ [
0, 1 ] n , we can efficiently determine the index of a simplex that contains itas follows. First find the base y of a cube of U nm that contains x . Next, find a permutation π such that x π ( ) − y π ( ) ≥ · · · ≥ x π ( n ) − y π ( n ) . Then, it follows that ( y , π ) is the index of asimplex containing x . Thus, the value function can be computed efficiently.• Given an n -simplex { z , . . . , z n } and i ∈ {
0, 1, . . . , n } , we can efficiently compute the index ofthe other n -simplex that also has { z , . . . , z n } \ { z i } as a facet. This is called a pivot operation. A.2 The Borsuk-Ulam Theorem and Tucker’s Lemma
Here, we provide the definitions of the problems that we generalize. Note that in Section 3, weexplain how the Borsuk-Ulam Theorem can be interpreted under a more general definition, whichexplains how our definition of Polygon Borsuk-Ulam is indeed a generalization. We start withthe Borsuk-Ulam Theorem, which is usually stated as “for every continuous function from S n to R n , there exists a point x ∈ R n , such that f ( x ) = f ( − x ) ”. We present an alternative definition (asstated in [Matoušek, 2008]), which is more appropriate for our results.57 heorem A.1 (B orsuk -U lam T heorem [Borsuk, 1933, Matoušek, 2008]) . For every antipodal mappingf : S n → R n (i.e., a function f which is continuous and f ( x ) = f ( − x ) ), there exists a point x ∈ S n suchthat f ( x ) = . Tucker’s lemma is a well-known combinatorial existence theorem, which is an analogue ofthe Borsuk-Ulam Theorem. It is usually stated on the d -dimensional unit ball or sphere. Forcomputational purposes the following version is more commonly used. Theorem A.2 (T ucker ’ s lemma [Tucker, 1945]) . Let m , d ≥ . Let λ : ([ − m , m ] ∩ N ) d →{± ±
2, . . . , ± d } be any labeling that satisfies λ ( − x ) = − λ ( x ) for all x ∈ ([ − m , m ] ∩ N ) d suchthat ∃ i with | x i | = m. Then there exist two points x , y ∈ ([ − m , m ] ∩ N ) d with (cid:107) x − y (cid:107) ∞ ≤ that haveopposite labels, i.e., λ ( x ) = − λ ( y ) . The corresponding computational problem T ucker is known to be PPA-complete [Papadim-itriou, 1994, Aisenberg et al., 2020], even if the dimension is fixed to be d = A.3 Z p -equivariant tie-breaking For some of our constructions, we will require a tie-breaking that is Z p -equivariant and efficientlycomputable. Observe that non-efficient tie breaking rules exist by carefully choosing one represen-tative for any equivalence class but this is not sufficient for our proofs. We define an efficientlycomputable rule below for any set S ∈ Z p \ { ∅ , Z p } , let S + i : = { x + i : x ∈ S } . Definition 19 ( Z p - equivariant tie - breaking ) . For any prime p , the Z p -equivariant tie-breakingfunction T p : 2 Z p \ { ∅ , Z p } → Z p is computed as follows on input S ∈ Z p \ { ∅ , Z p } :1. For every i ∈ Z p , write S + i as a bit-string of length p . Namely, construct the bit-string b ( i ) where the j th bit from the left indicates whether j ∈ S + i (for j =
0, . . . , p − − i ∗ ∈ Z p , where i ∗ = argmax i b ( i ) ( b ( i ) interpreted as a number in binary). Lemma A.3.
For any prime p, the Z p -equivariant tie-breaking function T p : 2 Z p \ { ∅ , Z p } → Z p iswell-defined and satisfies for any S ∈ Z p \ { ∅ , Z p } : • T p ( S ) ∈ S • T p ( S + i ) = T p ( S ) + i for all i ∈ Z p .Proof. If p is prime and S + i = S for some i ∈ Z p \ { } , then S ∈ { ∅ , Z p } . This follows fromobserving that the corresponding bit strings b ( ) and b ( i ) must be equal and that this implies thatthe bits of b ( ) with index { k · i } k ∈ Z p are all equal. Since p is prime, { k · i } k ∈ Z p = Z p .Hence, |{ S + i : i ∈ Z p }| = p for any S ∈ Z p \ { ∅ , Z p } . Thus, the bit-strings b ( i ) are alldistinct, T p ( S ) is unique and T p is well-defined. Next, by construction, it is easy to see that T p ( S + i ) = − ( i ∗ − i ) = T p ( S ) + i . Finally, since S (cid:54) = ∅ , i ∗ will be such that b ( i ∗ ) has a 1 in theleft-most position. Thus, 0 ∈ S + i ∗ , which implies that − i ∗ ∈ S . Example.
Let p = S = { } . Then, S + = { } , S + = { } , S + = { } , and b ( ) = b ( ) =
001 and b ( ) =
100 (in binary). From Definition 19, T ( S ) = ( p , n ) -BSS-Tucker reduces to ( p , n + ) -BSS-Tucker Let ( p , n ) -BSS-T ucker denote the p -BSS-T ucker problem with dimension parameter n . We havethe following lemma. Lemma B.1.
For all n ≥ and prime p ≥ , ( p , n ) - BSS-T ucker reduces to ( p , n + ) - BSS-T ucker .Proof.
The domain of ( p , n ) -BSS-T ucker with Kuhn’s triangulation can be written as X n = { ( c , . . . , c p ) ∈ U npm |∀ j ∈ [ n ] , ∃ i ∈ [ p ] : c ij = } .Note that the subset of X n + corresponding to c n + = · · · = c pn + = X n . Since we use Kuhn’s triangulation in both cases, the triangulations “match”.Let λ be an instance of ( p , n ) -BSS-T ucker . We construct an instance λ (cid:48) of ( p , n + ) -BSS-T ucker as follows. For any vertex ( c , . . . , c p ) , we set λ (cid:48) ( c , . . . , c p ) = (cid:40) λ ( c , . . . , c p ) , if c n + = · · · = c pn + = ( k , n + ) where k = T p ( argmax i ∈ [ p ] c in + ) , otherwiseIn the second case, we have used the tie-breaking rule defined in Definition 19. Since thistie-breaking is Z p -equivariant, it is easy to see that λ (cid:48) also satisfies the boundary conditions.Consider any solution to this instance, i.e., a ( p − ) -simplex σ that has all labels ( (cid:96) ) , . . . , ( p , (cid:96) ) for some (cid:96) . If (cid:96) = n +
1, then there exists i ∈ [ p ] such that c in + = for all vertices of σ . Thisfollows from the fact that the triangulation is “nice”. But then, σ cannot have the label ( i , n + ) .Hence, it must be that (cid:96) < n + σ is contained in the region identified with X n where λ = λ (cid:48) .Thus, σ also yields a solution to the original instance. C Z p - star -T ucker is in PPA- p , Full Proof In this section, we provide the full proof of Theorem 5.3. Namely, we show how to reduce Z p - star -T ucker to I mbalance - mod - p . Proof.
The proof is a generalization of the construction given by Freund and Todd [1981] forTucker’s lemma. The main difficulties in generalizing their approach are:• For p =
2, when a path hits the boundary, there is a corresponding path that also hits theboundary on the antipodal side, and we can join the two endpoints. For p >
2, when a pathhits the boundary, there are now p − p =
2, the original construction associates a label with each axis of the domain. For p >
2, there are more axes than labels, and so a single label must be associated to multipleaxes. This creates imbalanced nodes in the graph that are not solutions. We solve thisproblem by carefully assigning weights to the edges of the graph.59 ub-orthants.
Recall that d = t ( p − ) . Consider the domain R dp , m with a labeling function λ given by a Boolean circuit. Let T be Kuhn’s triangulation of R dp , m as described earlier. The domain R dp , m can be subdivided into what we call sub-orthants, which are orthants of coordinate subspaces.Formally, a sub-orthant is a space of the form A × A × · · · × A d , where A (cid:96) = {∗ i (cid:96) j : 0 ≤ j ≤ m } or A (cid:96) = { } for (cid:96) =
1, . . . , d .We associate a label to every axis of R dp , m as follows. The label ∗ i j is associated to the ∗ i -axis ofthe [( j − )( p − ) + (cid:96) ] th copy of R p , m , for (cid:96) =
1, 2, . . . , p −
1. Thus, every label is associated toexactly p − X , let S ( X ) denote the set of labels associated with theaxes that are used by X . For any simplex σ of T , we let O ( σ ) denote the smallest sub-orthant thatcontains σ .For any sub-orthant O and j ∈ [ t ] , let r j ( O ) be the number of coordinates in the range ( j − )( p − ) +
1, . . . , j ( p − ) that are equal to 0 in O . In particular, we have ∑ tj = r j ( O ) = t ( p − ) − dim ( O ) . Note that if | S ( O ) | = dim ( O ) , then r j ( O ) = p − − |{ i ∈ Z p : ∗ i j ∈ S ( O ) }| .We abuse notation and denote r j ( σ ) : = r j ( O ( σ )) . Happy simplices.
Let k ∈ {
0, 1, . . . , d } . A k -dimensional simplex σ of T is happy , if1. dim ( O ( σ )) = k ( σ is full-dimensional in its sub-orthant) | S ( O ( σ )) | = dim ( O ( σ )) (the sub-orthant uses ≤ axis associated with each label) S ( O ( σ )) ⊆ λ ( σ ) ( σ carries all the labels associated with its sub-orthant) A happy simplex is called super-happy if we actually have S ( O ( σ )) (cid:40) λ ( σ ) . In particular, the0-dimensional simplex 0 d is super-happy.Consider a happy simplex σ that has a facet τ ⊂ ∂ R dp , m such that λ ( τ ) = S ( O ( σ )) . Such asimplex is called a boundary-happy simplex. If σ is such a simplex, then the simplex that has θτ as a facet is also boundary-happy. Thus, we group τ , θτ , . . . , θ p − τ together into an equivalenceclass [ τ ] . Every such equivalence class has size exactly p . Formally, let B be the set of all simplices τ ⊂ ∂ R dp , m such that λ ( τ ) = S ( O ( τ )) and | S ( O ( τ )) | = dim ( O ( τ )) . We define an equivalencerelation on B by τ ≡ τ if and only if ∃ i ∈ Z p such that τ = θ i τ .We construct a graph G . The vertices of G are the happy simplices of T and the equivalenceclasses [ τ ] (for all τ ∈ B ). Orientation.
We now define an orientation for our simplices. Fix an ordering of the labels, e.g., ∗ ∗
1, . . . , ∗ p ∗
2, . . . . Let σ be any happy k -simplex, k ≥
1, and let x x . . . x k be any orderingof its vertices. Let (cid:98) x , (cid:98) x , . . . , (cid:98) x k ∈ N k denote the coordinates of x , x , . . . , x k in O ( σ ) , where thecoordinates are ordered according to the fixed ordering of the labels. Note that there is at mostone coordinate associated to each label (because | S ( O ( σ )) | = k ) and thus the coordinate vectorsare uniquely determined. Furthermore, note that the coordinates are non-negative. We define the k × k matrix M = [ (cid:98) x − (cid:98) x , (cid:98) x − (cid:98) x , . . . , (cid:98) x − (cid:98) x k ] , i.e., the i th column is (cid:98) x − (cid:98) x i . Then, we define theorientation of happy simplex σ with ordering x x . . . x k as or ( σ | x . . . x k ) = det M . Note that byconstruction of the triangulation T , we always have det M ∈ {− + } . Indeed, it is easy to checkthat M can be transformed into the identity matrix by elementary operations that can only changethe sign of the determinant. Edges.
Let σ be a happy k -simplex, k ≥
1. If σ is super-happy, then it has a single facet τ suchthat λ ( τ ) = S ( O ( σ )) . If it is not super-happy, then it has exactly two facets τ and τ such that λ ( τ i ) = S ( O ( σ )) , i =
1, 2. In any case, any such facet τ of σ yields an edge as follows:60 If τ does not lie in the boundary of the sub-orthant O ( σ ) , then there is exactly one other k -simplex σ (cid:48) in O ( σ ) that also has τ as its facet. σ (cid:48) is also happy, and we put an edge between σ and σ (cid:48) .• If τ lies in the boundary of the sub-orthant O ( σ ) , there are two cases: – τ lies in ∂ R dp , m . In that case, σ is a boundary-happy simplex and we put an edgebetween σ and [ τ ] . – τ does not lie in ∂ R dp , m . Then, τ is a super-happy ( k − ) -simplex and we put an edgebetween σ and τ .In all of these cases, the direction of the edge is determined as follows. Let x x . . . x k be theordering of the vertices of σ such that τ = { x , . . . , x k } and x . . . x k are ordered according to theirlabels. If or ( σ | x . . . x k ) =
1, then the edge is incoming into σ . Otherwise, it is outgoing out of σ .Finally, the weight of the edge is always ∏ tj = r j ( σ ) !.By the definition, it follows that there are three types of edges:• (Type 1) An edge between two happy k -simplices σ , σ that lie in the same sub-orthant andshare a facet τ with λ ( τ ) = S ( O ( σ i )) , i =
1, 2.• (Type 2) An edge between a happy simplex σ and its super-happy facet τ such that λ ( τ ) = S ( O ( σ )) .• (Type 3) An edge between a boundary-happy simplex σ and its facet equivalence class [ τ ] .An edge of Type 2 or 3 is always “created” by exactly one of its endpoints. Thus, its directionand weight are well-defined. An edge of Type 1 however, is “created” by both of its endpointsand we will prove that it is well-defined, i.e., both endpoints agree on its direction and weight.For the weight this is easy to see, since O ( σ ) = O ( σ ) implies that r j ( σ ) = r j ( σ ) for all j . Forthe direction, we postpone this consistency check to the end of the proof.We now prove that all vertices of G that do not yield a solution are balanced modulo p , except0 d . The trivial solution d . We have λ ( d ) = ∗ i j . There are exactly p − O such that S ( O ) = {∗ i j } and each of them contains a happy 1-simplex σ that has 0 d as a facet. It followsthat 0 d has p − (( p − ) ! ) t / ( p − ) . Furthermore, all the edges areoutgoing, because or ( σ | x d ) = σ ). It follows that the total imbalance of0 d is ( p − )(( p − ) ! ) t / ( p − ) = (( p − ) ! ) t = ( − ) t mod p , where we used ( p − ) ! = − p since p is prime (Wilson’s theorem). It follows that 0 d is always a valid trivial solution forI mbalance - mod - p , because ( − ) t (cid:54) = p for all t ≥ Happy, but not super-happy.
Consider a happy k -simplex σ with two facets τ , τ that satisfy λ ( τ i ) = S ( O ( σ )) , i =
1, 2. Then, σ has two edges and they both have the same weight. Let x x . . . x k be the ordering of σ such that τ = { x , . . . , x k } and x . . . x k are ordered accordingto their labels. Let i ∈ [ k ] be the index such that x i and x have the same label. In particular, τ = { x , . . . , x i − , x , x i + , . . . , x k } and x . . . x i − x x i + . . . x k are ordered according to their labels.We havedet [ (cid:98) x − (cid:98) x , (cid:98) x − (cid:98) x , . . . , (cid:98) x − (cid:98) x k ] = det [ (cid:98) x i − (cid:98) x , . . . , (cid:98) x i − (cid:98) x i − , (cid:98) x − (cid:98) x i , (cid:98) x i − (cid:98) x i + , . . . , (cid:98) x i − (cid:98) x k ]= − det [ (cid:98) x i − (cid:98) x , . . . , (cid:98) x i − (cid:98) x i − , (cid:98) x i − (cid:98) x , (cid:98) x i − (cid:98) x i + , . . . , (cid:98) x i − (cid:98) x k ] i th column from all other columns, and then multiplied the i thcolumn by −
1. It follows that or ( σ | x . . . x k ) = − or ( σ | x i x . . . x i − x x i + . . . x k ) . Thus, one edge isincoming and the other outgoing, i.e., σ is balanced. Equivalence class.
Consider an equivalence class [ τ ] . Let σ be the happy k -simplex that has τ asa facet. In G , [ τ ] has exactly p edges: one with each of σ , θσ , θ σ , . . . , θ p − σ . Since S ( O ( θ i σ )) = θ i S ( O ( σ )) for all i , it follows that r j ( θ i σ ) = r j ( σ ) for all i , j . Thus, all p edges have the same weight.Let x . . . x k be the ordering of σ such that τ = { x , . . . , x k } and x . . . x k are ordered according totheir labels. Let y i = θ x i for all i . Then, y . . . y k might not be ordered according to their labels.We let π denote the permutation that we would have to apply to order them correctly. As before, (cid:98) x i denotes the coordinates of x i restricted to O ( σ ) , where the coordinates are ordered according tothe associated label. (cid:98) y i denotes the coordinates of y i restricted to O ( θσ ) , where the coordinates areordered according to the associated label. Since the associated labels have changed according to θ ,it follows that if we re-order the coordinates of (cid:98) x i according to π we obtain (cid:98) y i for all i =
0, 1, . . . , k .Thus, we have or ( θσ | y π ( y . . . y k )) = sgn ( π ) det [ (cid:98) y − (cid:98) y , (cid:98) y − (cid:98) y , . . . , (cid:98) y − (cid:98) y k ]= sgn ( π ) [ (cid:98) x − (cid:98) x , (cid:98) x − (cid:98) x , . . . , (cid:98) x − (cid:98) x k ]= or ( σ | x . . . x k ) It follows that all edges of [ τ ] are directed the same way, i.e., they are all incoming or all outgoing.Since there are p edges and they also have the same weight, it follows that [ τ ] has imbalance 0modulo p . In this argument we assumed that λ satisfies the boundary conditions. Thus, if [ τ ] isnot balanced modulo p , we obtain a counter-example, which is a solution. Super-happy.
Consider a super-happy k -simplex σ , k ≥
1. Note that σ has a single facet τ thatsatisfies λ ( τ ) = S ( O ( σ )) . Thus, σ “creates” a single edge. Let x . . . x k be the ordering of σ suchthat τ = { x , . . . , x k } and x . . . x k are ordered according to their labels. The edge has weight ∏ tj = r j ( σ ) ! and it is incoming if or ( σ | x . . . x k ) =
1, outgoing otherwise.Since σ is super-happy, we have λ ( x ) / ∈ λ ( τ ) = S ( O ( σ )) . Let ∗ i (cid:96) = λ ( x ) . If {∗ j (cid:96) | j ∈ [ p ] \ { i }} ⊆ S ( O ( σ )) , or equivalently if r (cid:96) ( σ ) =
0, then σ yields a solution. Otherwise, thereare exactly r (cid:96) ( σ ) different sub-orthants O such that O ( σ ) ⊂ O and S ( O ) = S ( O ( σ )) ∪ {∗ i (cid:96) } .Thus, there are exactly r (cid:96) ( σ ) happy ( k + ) -simplices ρ such that σ is a facet of ρ and ρ is happybecause of σ (i.e., S ( O ( ρ )) = λ ( σ ) ). It follows that σ has r (cid:96) ( σ ) additional edges (apart fromthe one it “created”). Each of these edges has weight ∏ tj = r j ( ρ ) ! = ( r (cid:96) ( σ )) − ∏ tj = r j ( σ ) !, since r (cid:96) ( ρ ) = r (cid:96) ( σ ) −
1. Thus, if these r (cid:96) ( σ ) edges have opposite direction to the edge “created” by σ (from the perspective of σ ), σ will be balanced.Consider any such ρ . Let x . . . x k + be the ordering of ρ such that σ = { x , . . . , x k + } and x . . . x k + are ordered according to their labels. Let t ∈ [ k + ] be the index of label ∗ i (cid:96) if we orderthe labels in S ( O ( ρ )) . Then, we also have that τ = σ \ { x t } . Furthermore, x . . . x t − x t + . . . x k + are also ordered according to their labels. From the perspective of σ , the edge it “created” isdirected according to or ( σ | x t x . . . x t − x t + . . . x k + ) and the edge created by ρ is directed accordingto − or ( ρ | x x . . . x k + ) . As before, let (cid:98) x j denote the coordinates of x j restricted to O ( ρ ) , wherethe coordinates are ordered according to the associated label. Let ¯ x j denote the coordinates of x j restricted to O ( σ ) , where the coordinates are ordered according to the associated label. Note that62f we remove the t th coordinate from (cid:98) x j , then we obtain ¯ x j . We now haveor ( ρ | x x . . . x k + ) = det [ (cid:98) x − (cid:98) x , . . . , (cid:98) x − (cid:98) x k + ]= det [ (cid:98) x t − (cid:98) x , . . . , (cid:98) x t − (cid:98) x t − , (cid:98) x − (cid:98) x t , (cid:98) x t − (cid:98) x t + , . . . , (cid:98) x t − (cid:98) x k + ]= ( − ) t + t det [ ¯ x t − ¯ x , . . . , ¯ x t − ¯ x t − , ¯ x t − ¯ x t + , . . . , ¯ x t − ¯ x k + ]= or ( σ | x t x . . . x t − x t + . . . x k + ) where we first subtracted the t th column from all other columns, and then we used Laplace’sdeterminant formula along the t th row. Note that the t th entry in (cid:98) x t − (cid:98) x j is 0 for all j ∈ [ k + ] and it is 1 in (cid:98) x − (cid:98) x t . Consistency.
The only thing that remains to be checked is that edges of Type 1 are well-defined,in terms of the direction. Let σ , σ be two happy k -simplices that lie in the same sub-orthantand share a facet τ with λ ( τ ) = S ( O ( σ i )) , i =
1, 2. Let { x , . . . , x k } = τ and x . . . x k be theordering according to their labels. Let { x } = σ \ τ and { x (cid:48) } = σ \ τ . We want to showthat or ( σ | x x . . . x k ) and or ( σ | x (cid:48) x . . . x k ) have opposite signs. This can be proved directlycombinatorially by using the way the triangulation is constructed, but we provide a proof that ismore general here. Let φ : R k → R k be the unique linear function such that φ ( (cid:98) x − (cid:98) x ) = (cid:98) x (cid:48) − (cid:98) x and φ ( (cid:98) x − (cid:98) x i ) = (cid:98) x − (cid:98) x i for all i ∈ [ k ] \ { } . φ is unique, because (cid:98) x − (cid:98) x , (cid:98) x − (cid:98) x , . . . , (cid:98) x − (cid:98) x k form a basis of R k . φ is the identity function on the hyperplane given by (cid:98) x − (cid:98) x , . . . , (cid:98) x − (cid:98) x k andmaps (cid:98) x − (cid:98) x to (cid:98) x (cid:48) − (cid:98) x . Since x and x (cid:48) lie on opposite sides of the hyperplane defined by τ , itfollows that (cid:98) x − (cid:98) x and (cid:98) x (cid:48) − (cid:98) x lie on opposite sides of the hyperplane on which φ is the identity.It follows that det φ <
0. Thus, we can writedet [ (cid:98) x (cid:48) − (cid:98) x , . . . , (cid:98) x (cid:48) − (cid:98) x k ] = det [ (cid:98) x (cid:48) − (cid:98) x , (cid:98) x − (cid:98) x . . . , (cid:98) x − (cid:98) x k ]= det φ det [ (cid:98) x − (cid:98) x , (cid:98) x − (cid:98) x . . . , (cid:98) x − (cid:98) x k ]= det φ det [ (cid:98) x − (cid:98) x , . . . , (cid:98) x − (cid:98) x k ]]