A New Combinatorial Property of Geometric Unique Sink Orientations
AA New Combinatorial Property of Geometric Unique SinkOrientations
Yuan Gao ∗ Bernd G¨artner † Jourdain Lamperski ‡ August 21, 2020
Abstract
A unique sink orientation (USO) is an orientation of the hypercube graph with the prop-erty that every face has a unique sink. A number of well-studied problems reduce in stronglypolynomial time to finding the global sink of a USO; most notably, linear programming (LP)and the P-matrix linear complementarity problem (P-LCP). The former is not known tohave a strongly polynomial-time algorithm, while the latter is not known to even have apolynomial-time algorithm, motivating the problem to find the global sink of a USO. Al-though, every known class of geometric
USOs, arising from a concrete problem such as LP, isexponentially small, relative to the class of all USOs. Accordingly, geometric USOs exhibitadditional properties that set them apart from general USOs, and it may be advantageous,if not necessary, to leverage these properties to find the global sink of a USO faster. Only afew such properties are known. In this paper, we establish a new combinatorial property ofthe USOs that arise from symmetric
P-LCP, which includes the USOs that arise from linearand simple convex quadratic programming. A unique sink orientation (USO) is an orientation of the n -dimensional hypercube graph (the n -cube) with the property that every face (subcube) has a unique sink. Figure 1 depicts a 3-dimensional USO called the spinner and highlights the unique sink of the whole cube (which isitself a face) and the unique sink of each of the six 2-faces.USOs were introduced by Stickney and Watson as a combinatorial model to study principalpivoting algorithms for P-matrix linear complementarity problems (P-LCPs) [29]. The USOsresulting from P-LCPs are called P-cubes . The spinner in Figure 1 is actually Stickney andWatson’s digraph 19 .After having been forgotten for more than twenty years, USOs were rediscovered by Sz´abo andWelzl [30], motivated by applications in optimization theory, but mostly by their simple and cleancombinatorial structure. ∗ Institute of Theoretical Computer Science, Department of Computer Science, ETH Zurich, Zurich, Switzerland † Institute of Theoretical Computer Science, Department of Computer Science, ETH Zurich, Zurich, Switzerland ‡ Operations Research Center, Massachusetts Institute of Technology, Cambridge MA, USA a r X i v : . [ m a t h . C O ] A ug igure 1: The spinner, a 3-dimensional USO; the black circle is the unique global sink, and the redcircles are the unique sinks in the six 2-faces. The USO is called the spinner because it containsa directed cycle of length 6 and is rotationally symmetric under any cyclic permutation of thedimensions. The algorithmic problem.
The connection to optimization is as follows. The unique global sink S corresponds to an optimal solution, among a set of candidate solutions (the 2 n cube vertices),to some optimization problem. The USO defines a structure on the search space that is notexplicitly given (because of its exponential size) but can be queried locally. That is, we have a vertex evaluation oracle that takes as input any vertex V and returns the orientations of the n edges incident to V . The algorithmic question is: how many vertex evaluations do we need to find S ? In applications, vertex evaluation can typically be performed in polynomial time.Clearly we can find the global sink with 2 n vertex evaluations; however, we can do better. Priorto Sz´abo and Welzl, a number of acyclic combinatorial models had been studied. These modelsarise when candidate solutions are ordered by an objective function. In such models, subexponential randomized algorithms exist [20, 10]; the main ideas involved can essentially be distilled into the Random Facet algorithm, which requires an expected number of at most e √ n vertex evaluations tofind the global sink of an acyclic USO [13]. However, USOs may in general contain directed cycles(see Figure 1 for the smallest example); in particular, P-cubes may contain directed cycles. Sz´aboand Welzl were the first to come up with nontrivial algorithms for this more general context. Theydeveloped a deterministic algorithm that uses at most 1 . n vertex evaluations [30, Theorem4.1] and a randomized algorithm [30, Lemma 3.2] that uses at most 1 . n vertex evaluations inexpectation when combined with the optimal randomized algorithm for the 3-cube [31].The aforementioned algorithmic results have not be improved during the last twenty years,although there seems to be a lot of room for improvement. Indeed, the best known lower boundfor the number of vertex evaluations is a small polynomial Ω( n / log n ) [25]. Geometric USOs.
It might be that USOs are algorithmically difficult because they form anextremely rich class of orientations that contains many complicated orientations that we neverencounter “in practice”. Let us elaborate on this. As the n -cube has n n − edges, the number of n -cube orientations is 2 n n − . The number of USOs is somewhat smaller but still doubly exponential,namely 2 Θ(2 n log n ) [19]. In contrast, the number of USOs that we encounter in applications istiny, as we explain next. Application areas include linear complementarity (the original source ofUSOs [29]), linear programming [12, 13], and computational geometry [7]. In all of these areas,a USO is typically defined by polynomially (in n ) many input numbers; for example, a matrix M ∈ R n × n and a vector q ∈ R n , in the case of P-cubes. More precisely, the edge orientations aredetermined by signs of algebraic expressions (all of polynomial degree) over the input numbers.In this situation, the number of USOs that can be obtained from all possible inputs is singly2xponential. For example, there are only 2 Θ( n ) P-cubes [9]. Let us say that a geometric
USO isone that is generated from an input of polynomial size, as previously described.Hence, the number of geometric USOs (the ones that we actually want to work with) is ex-ponentially smaller than the number of all USOs. It is therefore natural to ask whether we canexploit this fact algorithmically. That is, why bother with all USOs if we are only interested in atiny fraction of them? It is also natural to ask if the USO abstraction is even worth consideringgiven that it significantly overestimates the number of problem instances. There is an importantreason for the abstraction: a simple combinatorial model might allow us to see and exploit relevantstructure of a concrete problem that otherwise would remain hidden behind the input numbers. Astriking example of this is the subexponential algorithm (mentioned before) in the combinatorialmodel of
LP-type problems [26, 20]. Although developed in a completely abstract setting, it wasat the same time a breakthrough result in the theory of linear programming.On the other hand, a combinatorial model typically fails to capture some properties of theunderlying geometric situation and therefore loses information. A well-known example is Pappus’stheorem that no longer holds when we move from a geometric setting (vector configurations) to acombinatorial setting (matroids) [22, Proposition 6.1.10]. Still, we might be able to regain someinformation by extracting additional combinatorial structure from the geometric situation. In theremainder of this introduction, we review the known results in this direction for geometric USOsand introduce the new combinatorial property that we establish in this paper.
P-cubes satisfy the Holt-Klee property.
A classical example of a combinatorial model isa polytope graph , a graph formed by the vertices and edges of a polytope. It turns out thatmany questions about polytopes can be answered through a sufficient understanding of polytopegraphs and their properties. An early related result is Balinski’s theorem: the graph of an n -dimensional polytope is n -connected [1]. Holt and Klee later established a directed version ofBalinski’s theorem: in the graph of an n -dimensional polytope, with edges oriented according toa generic linear function, there are n directed paths from the highest to the lowest vertex thatare internally disjoint [16]. The Holt-Klee property generalizes from geometric cubes to P-cubes.For P-cubes, it states that there are n directed paths from the (unique) source to the unique sinkthat are internally disjoint [14, Corollary 4.4]. Figure 2 shows such paths for the spinner (which isactually a P-cube; see Section 4), along with a USO that fails to satisfy the Holt-Klee property.Figure 2: The spinner on the left satisfies the Holt-Klee property because there are 3 internallydisjoint directed paths from the global source to the global sink. The USO on the right does notsatisfy the Holt-Klee property because the red and the blue path necessarily interfere.A USO that satisfies the Holt-Klee property is not necessarily a P-cube; the Holt-Klee propertyis necessary but not sufficient. In fact, it dramatically fails to be sufficient because the number3f USOs that satisfy the Holty-Klee property is again doubly exponential [9]. Although, it mightbe the case that such USOs, despite the large number of them, are algorithmically easier thangeneral USOs. But so far, there is no real indication of this. The only known result is as follows.Matouˇsek showed that the Random Facet algorithm may actually need an expected number of e Θ( √ n ) vertex evaluations for a class of (acyclic) USOs [18, 13]. The lower bound is attained fora random member of the class. In contrast, all members of the class with the Holt-Klee propertyare solved by Random Facet in expected O ( n ) vertex evaluations [13]. K-cubes are locally uniform.
K-cubes are the USOs that arise from K-matrix linear comple-mentarity problems (K-LCPs). K-LCPs form an “easy” subclass of P-LCPs that have long knownto be solvable in polynomial time (for P-LCPs, this is open) [3, 23]. K-cubes were shown to be locally uniform [8]. Identifying the cube vertices with the subsets of [ n ], this means the following:if some vertex V has only incoming edges from (or only outgoing edges to) k “higher” neighbors V ∪{ i } , . . . , V ∪{ i k } , then all edges in the k -face spanned by these vertices have the same direction(all down, or all up); see Figure 3. ⇒ V V ∪ { i } V ∪ { i } V ∪ { i } Figure 3: Locally uniform USOs: whenever the lowest vertex V in a face has only incoming (oronly outgoing) edges to its neighbors in the face, then all edges in the face point downwards (orupwards).This property can in fact be exploited algorithmically: in a locally uniform USO, every directedpath has length at most 2 n [8, Theorem 5.9]. Hence, we can find the sink with a linear number ofvertex evaluations, starting at any vertex, and simply following a directed path. When applied toK-cubes, this simply enhances the arsenal of available polynomial-time methods for K-LCP, butthe number of locally uniform USOs is doubly exponential and therefore much larger than thenumber of K-cubes [9]. Tridiagonal and Hessenberg P-LCPs.
A matrix M = ( m ij ) ∈ R n × n is tridiagonal if m ij = 0whenever | i − j | >
1, and more generally (lower)
Hessenberg if m ij = 0 whenever j − i >
1. P-LCPs with Hessenberg matrices can be solved in polynomial time [15]. Although this can be shownwithout considering USOs; the main idea involved is a combinatorial property of the correspondingUSOs [28].
Our contribution: D-cubes satisfy property L.
There is another and actually quite in-teresting subclass of P-LCPs, namely the ones with a symmetric P-matrix. As the symmetricP-matrices are exactly the positive d efinite matrices, we refer to the USOs generated by symmet-ric P-LCPs as D-cubes . Unlike general P-LCPs, symmetric P-LCPs are equivalent to a class of4strongly) convex quadratic programs [5, Section 1.2] and as such can be solved in polynomial timeusing the ellipsoid method [17]. Still, unlike K-LCPs, the symmtric P-LCPs cannot be called an“easy” subclass of all P-LCPs. In particular, no strongly polynomial-time algorithms are known;these are algorithms with a number of arithmetic operations that can be polynomially bounded in n . This means, the situation is the same as for linear programming, where the problem of findinga strongly polynomial-time algorithm is on Steve Smale’s 1998 list of mathematical problems forthe next century [27].Promising candidates for strongly polynomial-time algorithms are combinatorial methods that—unlike the ellipsoid method—explore a discrete search space, where every search step can be im-plemented in strongly polynomial time. The most prominent combinatorial method for linearprogramming is the simplex method. But as there is a (strongly polynomial-time) reduction fromlinear programming to the problem of finding the sink in a D-cube [12], methods for the latterproblem are also relevant for the former. In fact, the fastest deterministic combinatorial algorithm(currently known) for solving linear programs with 2 n constraints and n variables is based on thisreduction [12].The new combinatorial property of D-cubes that we establish is as follows. For every vertex V in a USO, and for all i, j / ∈ V , consider the 2-face spanned by V, V ∪ { i } , V ∪ { j } , V ∪ { i, j } ;see Figure 4. We write i → j if in this face, the edges along dimension j have opposite directions.As the face has a unique sink, it is easy to see that we cannot simultaneously have j → i . But wemay have neither of the two directions. V V ∪ { i } V ∪ { j } V ∪ { i, j } V V ∪ { i } V ∪ { j } V ∪ { i, j } j → i V V ∪ { i } V ∪ { j } V ∪ { i, j } i → j j (cid:54)→ ii (cid:54)→ ji (cid:54)→ jj (cid:54)→ i Figure 4: Property L at vertex V of a USO.This defines a directed graph on the complement of V , the L-graph of V , and we say that aUSO has property L if for each of its vertices V , the L-graph of V is acyclic. Our main result isthat every D-cube has property L. For example, this implies that the spinner in Figure 1, despitebeing a P-cube, cannot be a D-cube, as property L is violated at ∅ (the lower-left vertex). Wealso show that the number of USOs having property L is doubly exponential, so once more, thecombinatorial property is far from characterizing the geometric situation.The paper is organized as follows. In Section 2, we formally introduce cube orientations and theL-graphs of such orientations. Section 3 reviews USOs and shows that if a cube orientation satisfiesproperty L, it is already a USO. This may come as a surprise since property L only involves smalllocal graphs, whereas the USO property involves (exponentially large) faces. We also establisha doubly exponential lower bound on the number of USOs with property L. In Section 4, weintroduce P-cubes and in particular review their construction from P-LCPs, due to Stickney andWatson [29]. Section 5 contains our main result, a proof that every D-cube has property L. InSection 6, we show that, starting from dimension 4, there are USOs with no isomorphic copysatisfying property L, and we provide systematic constructions of such USOs. This shows thatproperty L is combinatorially nontrivial : it cannot always be attained by simply relabeling cube5ertices. But in the 3-dimensional case, an isomorphic copy with property L can always be found.Section 7 provides open problems and further research questions. We closely follow the notation of Sz´abo and Welzl [30]. For sets
U, V , define the symmetricdifference U ⊕ V = ( U ∪ V ) \ ( U ∩ V ). For U ⊆ W , define the interval [ U, W ] = { V : U ⊆ V ⊆ W } .A cube orientation is a directed graph O with vertex set vert O = [ U, W ], for some interval[
U, W ], that contains exactly one of the directed edges (
V, V ⊕ { i } ) and ( V ⊕ { i } , V ) for every V ∈ vert O and every i ∈ carr O := W \ U (the carrier of O ). We also write V → O V (cid:48) if O containsthe directed edge ( V, V (cid:48) ), or simply V → V (cid:48) if O is clear from the context. The dimension of O is | carr O| . We call O an n -cube orientation if vert O = [ ∅ , [ n ]], where [ n ] := { , , . . . , n } .The outmap of O is the function φ O : vert O → carr O defined by φ O ( V ) = { i ∈ carr O : V → O V ⊕ { i }} , V ∈ vert O . Hence, the set φ O ( V ) contains the cube dimensions along which V has outgoing edges in O .Next we define the central objects of this paper. Definition 1.
Let O be a cube orientation. For each vertex V ∈ vert O = [ U, W ] , the L-graphof V , denoted by L O ( V ) , is the directed graph with vertex set W \ V , and with an arc ( i, j ) for i, j ∈ W \ V, i (cid:54) = j , whenever j ∈ φ O ( V ) ⊕ φ O ( V ∪ { i } ) . (1)In words, the L-graph of V contains the arc ( i, j ) if exactly one of V and V ∪{ i } has an outgoingedge along dimension j ; see Figure 5. Note that the term “directed edge” is used in connectionwith cube orientations, while “arc” is used in connection with L-graphs. V V ∪ { i } V ∪ { j } V ∪ { i, j } ( i, j ) is an arc of L O ( V ) V V ∪ { i } V ∪ { j } V ∪ { i, j } ( i, j ) is not an arc of L O ( V ) i ijj Figure 5: Definition of the L-graph.The graph L O ( W ) is empty, and all graphs of the form L O ( W \ { i } ) are a single vertex. Hence,L-graphs are interesting only when | V | ≤ | W | −
2. Figure 6 shows a 4-cube orientation and fourof its L-graphs. In later figures, we draw the arcs of the L-graphs as red (double) arrows betweenthe corresponding dimensions of the cube orientation instead of explicitly drawing the L-graphs;see Figure 7.The arc(s) connecting i and j in L O ( V ) are determined by a 2-dimensional cube orientation.Up to isomorphism, there are four different such orientations: the eye , the bow , the twin peak , andthe cycle ; see Figure 7. The terms eye and bow are due to Szab´o and Welzl [30, Figure 4].6
23 4 23 4 23 3
123 4 L O ( ∅ ) L O ( { } ) L O ( { , } ) L O ( { , , } ) Figure 6: A 4-cube orientation and four of its 16 L-graphs.For an eye, there are no arcs between i and j . For a bow, there is exactly one of the arc, andfor the twin peak and the cycle, both arcs are present.Figure 8 depicts two 3-cube orientations and their (nontrivial) L-graphs of V = ∅ , { } , { } , { } .We see here that the left one, the spinner, has a cyclic L-graph, while the right one does not. Thismay be surprising, given that the two orientations are isomorphic. L-graphs have a “sense ofdirection”: for every vertex V , we only use the “higher-dimensions” i / ∈ V to define L O ( V ). Underan automorphism, a higher-dimension at V may become a lower-dimension in the isomorphic imageof V , and vice versa. In fact, the automorphism in Figure 8 precisely flips the higher-lower statusof dimension 1 at all vertices.Here is the central definition of this paper: Definition 2. An n -cube orientation O has property L if all of its L-graphs are acyclic. V V ∪ { i } V ∪ { j } V ∪ { i, j } V V ∪ { i } V ∪ { j } V ∪ { i, j } V V ∪ { i } V ∪ { j } V ∪ { i, j } V V ∪ { i } V ∪ { j } V ∪ { i, j } Eye Bow Twin Peak Cycle( i, j ) ( j, i )( i, j ) ( i, j )( j, i ) j j j ji i i i Figure 7: Possible arcs between i and j in L O ( V ).7 ∅{ } { }{ } { }{ } { } Figure 8: L-graphs of two isomorphic 3-cube orientations may not be isomorphic.Consider the 2-cube orientations of Figure 7 with V = ∅ , i = 1 , j = 2. The eye and thebow have property L. For the twin peak and the cycle, the L-graphs of ∅ each contain a cycle1 → →
1, so the twin peak and the cycle do not satisfy property L. Now consider Figure 8. The3-cube orientation on the right satisfies property L, but the left one, the spinner, does not becausethere is a cycle 1 → → → ∅ .Property L turns out to be invariant under the reversal of the direction of all arcs along a givenset R of dimensions. Let us formally introduce this operation, as we will need it later. Definition 3.
For a cube orientation O and R ⊆ carr O , let O ⊕ R be the cube orientation O (cid:48) with vert O (cid:48) = vert O and outmap φ O (cid:48) ( V ) = φ O ( V ) ⊕ R, V ∈ vert O (cid:48) . (2)In Figure 7, we can see this in action: the twin peak and the cycle can be obtained from eachother by reversing the edges along the vertical dimension. In contrast, the isomorphism betweenthe two 3-cube orientations in Figure 8 is not of this type: it does reverse all edges in dimension1, but on top of that, it also flips the two sides of the cube along this dimension. Observation 4.
Let O be a cube orientation, R ⊆ carr O , and O (cid:48) = O⊕ R . Then O has property Lif and only if O (cid:48) has property L.Proof. For all V ∈ vert O and i ∈ carr O , we have φ O ( V ) ⊕ φ O ( V ∪ { i } ) = φ O (cid:48) ( V ) ⊕ φ O (cid:48) ( V ∪ { i } )as a consequence of (2), so according to (1), O and O (cid:48) have the same L-graphs. We turn our attention to unique sink orientations (USOs), the subclass of cube orientations ofinterest. To define USOs, we need the concept of a face.
Definition 5. A face (or subcube) of a cube orientation O is a directed subgraph of O induced byan interval I ⊆ vert O . A face is proper if I (cid:54) = vert O . Faces are cube orientations themselves.
Definition 6.
A cube orientation O is a unique sink orientation (USO) if every face of O has aunique sink (a vertex with no outgoing edge).
8n particular, since the whole cube is itself a face, a USO has a unique global sink. Among thefour cube orientations depicted in Figure 7, the eye and the bow are USOs, while the two othersare not USOs. The twin peak has two global sinks, while the cycle has no global sink. (Faces ofdimension 0 (vertices) and 1 (edges) automatically have unique sinks.)The spinner in Figure 8 (left) is an example of a cyclic USO without property L. But reversing alledges along one of the dimensions yields an acyclic USO without property L, by Observation 4. Onthe other hand, the isomorphic copy of the spinner in Figure 8 (right) is cyclic but has property L.Hence, property L (acyclicity of all L-graphs) is at least not obviously related to acyclicity of theUSO itself.The first result of this paper is as follows.
Theorem 7.
If a cube orientation O has property L, then O is a USO. We have already “proved” Theorem 7 for the 2-dimensional case with Figure 7. Indeed, the2-dimensional cube orientations with property L (the eye and the bow) are USOs. The converseof Theorem 7 is false in dimension n ≥
3. For example, the spinner in Figure 1 is a 3-dimensionalUSO that does not have property L .For the proof of Theorem 7, we employ the concept of pseudo unique sink orientations (pseudoUSOs), which are minimal non-USOs. A pseudo USO is a cube orientation that does not have aunique global sink, but every proper face has a unique sink [2, Definition 2]. Every cube orientationthat is not a USO contains a face that is a m -dimensional pseudo USO, for some m ≥ Lemma 8.
Let O be a m -dimensional pseudo USO, m ≥ , with vert O = [ V, W ] and a globalsink at V . Then O contains a directed cycle among the set of vertices of the form V ∪ { i } and V ∪ { i, j } for i, j ∈ W \ V .Proof. Consider any vertex of the form V ∪ { i, j } . Since the proper face of O induced by theinterval [ V, V ∪ { i, j } ] = { V, V ∪ { i } , V ∪ { j } , V ∪ { i, j }} has a unique sink (namely V ), V ∪ { i, j } has an outgoing edge to one of V ∪ { i } and V ∪ { j } . W.l.o.g. suppose that it has an outgoingedge to V ∪ { i } . The vertex V ∪ { i } in turn has an outgoing edge to the global sink V , but sinceall outdegrees in O are even ( V has outdegree 0, and all outdegrees have the same parity), thereis another outgoing edge to some V ∪ { i, k } for k (cid:54) = j (the edge from V ∪ { i, j } was incoming).Now we repeat the argument from V ∪ { i, k } . Continuing in this way, we eventually construct adirected cycle. Proof of Theorem 7.
We show the contraposition: if O is not a USO, then it has a cyclic L-graph,so O fails to have property L. To this end, suppose that O is not a USO. Then O contains aface F that is a m -dimensional pseudo USO for some m ≥
2. Suppose that F is induced by theinterval [ V, W ]. Since reversing edges along any set of dimensions R neither affects property L(Observation 4) nor the pseudo USO property [2, Lemma 8], we may w.l.o.g. assume that V is aglobal sink of F . If m = 2, then F is a twin peak, and the L-graph L O ( V ) contains a directedcycle of length 2; see Figure 7. If m ≥
3, Lemma 8 yields the existence of a directed cycle V ∪ { i } → V ∪ { i , i } → V ∪ { i } → · · · → V ∪ { i (cid:96) } → V ∪ { i (cid:96) , i } → V ∪ { i } . in F . As V is the sink of F , the situation looks like Figure 9 for all t = 0 , . . . (cid:96) (we define i (cid:96) +1 = i ).Consequently, L O ( V ) contains all the arcs ( i t , i t +1 ) , t = 0 , . . . , (cid:96) and hence a directed cycle.9 V ∪ { i t } V ∪ { i t +1 } V ∪ { i t , i t +1 } ( i t , i t +1 ) i t i t +1 Figure 9: Proof that L O ( V ) contains a directed cycle.We conclude this section by showing that there are doubly exponentially many n -cube USOswith property L; hence, in “weeding out” non-geometric USOs, our new property L is not moreefficient than the previously known ones. Theorem 9.
There are at least n − n -cube USOs with property L.Proof. The bound is attained by the class of recursively combed n -cube USOs, which also providesthe best known asymptotic lower bound for the number of acyclic USOs [19]. A recursively combed n -cube USO O has all edges along dimension n oriented in the same way: either all of them go“up” ( V → V ∪ { n } ), or all of them go “down” ( V ∪ { n } → V ). Moreover, the two facets withcarrier [ n −
1] (“lower” and “upper” facet) are recursively combed USOs as well. For n = 1, everyUSO is recursively combed. It follows that the number r n of recursively combed n -cube USOssatisfies the recurrence relation r = 2 and r n = 2 r n − . This solves to r n = 2 n − .It is easy to see that all recursively combed USOs have property L. Indeed, since all directededges along dimension n have the same direction, no L-graph contains any arcs of the form i → n .As the arcs not involving n are contained in L-graphs of the lower or the upper facet, it inductivelyfollows that there are no arcs of the form i → j for i < j . Hence, all L-graphs are acyclic.The number of USOs is still significantly larger than 2 n − , namely n Θ(2 n ) [19]. To constructthis many USOs, we can start with the uniform USO ( V ∪ { i } → V for all V and all i / ∈ V ) andthen reverse all edges in a matching. The result is called a matching-reversal USO. The lowerbound then follows from a bound for the number of matchings in the n -cube [19]. We remarkthat this construction may yield USOs without property L; for example, the spinner in Figure 1is a matching-reversal USO. We do not know whether the lower bound in Theorem 9 can besignificantly improved. Stickney and Watson [29] were the first to show that every P-matrix linear complementarity prob-lem reduces to finding the global sink in a USO. In this section, we briefly review their construction.Given a matrix M ∈ R n × n and a vector q ∈ R n , the linear complementarity problem LCP( M, q )is to find vectors w , z ∈ R n such that w − M z = q , w , z ≥ , w (cid:62) z = 0 . While the first two conditions can be satisfied by solving a linear program, the third one makes theproblem hard in general. More precisely, it is NP-complete to decide whether there is a solution [4].10ut if M is a P-matrix (all principal minors are positive), the decision problem becomes trivial,because then there is a unique solution for every q [24]. The problem of finding the unique vectors w , z has unknown complexity status. It is unlikely to be NP-hard [21], but also no polynomial-timealgorithm is known. The problem falls into a number of more recent complexity classes, namelyPPAD, PLS, CLS, and UniqueEOPL [6], but it is not known to be complete for any of theseclasses.The reduction to USO is as follows. First, observe that w , z ≥ and w (cid:62) z = 0 together implythat for every i ∈ [ n ], one of w i and z i is zero. Suppose that for some V ⊆ [ n ], we set w i = 0 if i ∈ V and z i = 0 if i / ∈ V . Then there are unique values w i , for i / ∈ V and z i for i ∈ V such that w − M z = q . Indeed, taking the prescribed zeros into account, we must have − M V,V z V = q V ,where M V,V is the principal submatrix of M with rows and columns indexed by V , and z V is thesubvector of z with entries indexed by V .By definition of a P-matrix, M V,V has positive determinant and hence is invertible, so themissing (non-prescribed) z -entries z i for i ∈ V are uniquely determined. This in turn determinesthe missing w -entries via w = M z + q .Hence, the problem of solving LCP( M, q ) reduces to “guessing” a set S ⊆ [ n ] (an n -cubevertex) such that the missing entries of w and z are nonnegative.If q is generic (w.r.t. M ), meaning that no missing entry is 0, we can turn this guesswork intofinding the global sink in a suitably defined n -cube USO, with edge orientations defined by thesigns of the missing entries. If z and w (uniquely) solve w − M z = q ; w i = 0 , i ∈ V ; z i = 0 , i / ∈ V ,then we define V → V ⊕ { i } ⇔ z i < w i < . (3)To do this algebraically, we first formulate one system of equations that directly gives us themissing entries of w and z for a given V . When we write w − M z = q as I w − M z = q , where I is the ( n × n ) identity matrix, then we see that the n missing entries can be obtained by solving I V w V − M V z V = q , where V = [ n ] \ V and matrix subscripts select colums. With x i = z i if i ∈ V and x i = w i if i (cid:54)∈ V , this is equivalent to solving the system M ( V ) x = q , where the i -th column M ( V ) i of matrix M ( V ) is given by M ( V ) i = (cid:26) − M i , i ∈ V,I i , i / ∈ V. (4)As we know that the missing entries are uniquely determined, the matrix M ( V ) is invertiblefor all V . Hence, the outmap corresponding to the orientation in (3) is φ ( V ) = { i ∈ [ n ] : ( M ( V ) − q ) i < } , V ⊆ [ n ] . (5)Stickney and Watson have shown that φ is the outmap of an n -cube USO; its unique sink is the(unique) right guess for S [29]. Definition 10. A n -cube USO O is called a P-cube if its outmap is of the form given by (4) and(5) for some P-matrix M ∈ R n × n and generic q ∈ R n . For example, the spinner in Figure 1 is a P-cube, generated by M = , q = . (6)11 D-cubes
We now consider the subclass of P-cubes that arise from symmetric P -matrices. It turns out thatthese matrices are exactly the positive definite ones, the symmetric matrices such that x (cid:62) M x > x ∈ R n . The fact that a symmetric P-matrix is positive definite follows from Sylvester’scriterion , according to which a matrix is positive definite if and only if all its leading principalminors are positive. Since a symmetric P-matrix has all principal minors positive, it is positivedefinite. On the other hand, if we have a symmetric positive definite matrix, then in fact all principal minors are positive, so we have a symmetric P-matrix. Indeed, by permuting rows andcolumns in the same way, every principal minor can be made a leading principal minor of anothersymmetric positive definite matrix and is hence positive.
Definition 11. A n -cube USO O is called a D-cube if its outmap is of the form given by (4) and(5) for some symmetric positive definite matrix M ∈ R n × n and generic q ∈ R n . Here is the main result of the paper:
Theorem 12.
Every D-cube O has property L. This rules out that the spinner is a D-cube, but some isomorphic copies of it could still beD-cubes. Indeed, Stickney and Watson [29] have already shown that this is the case. With M = −
10 2 −
10 41 − − , q = − , we obtain the D-cube in Figure 10, an isomorphic copy of the spinner.Figure 10: A D-cube that is an isomorphic copy of the spinner (which is not a D-cube).Our main step towards the proof is Lemma 13 below. It generalizes an ad-hoc argument thatwas used to show that the spinner is not a D-cube [11]. Lemma 13. If O is a D-cube, then the L-graph L O ( ∅ ) is acyclic.Proof. Suppose that O is induced by M = ( m ij ) and q , where M is symmetric positive definite.Consider i, j ∈ [ n ] such that i (cid:54) = j . In O , we have ∅ → { i } ⇔ ( M ( ∅ ) − q ) i = q i < , { j } → { i, j } ⇔ ( M ( { j } ) − q ) i = q i − m ij m jj q j < , (7)12s a consequence of M ( { j } ) − = − m j . . . ... − m jj ... . . . − m nj − = − m j /m jj . . . ... − /m jj ... . . . − m nj /m jj . (8)Suppose that ( s, t ) is an arc in L O ( ∅ ), meaning that – by definition (1) of the local graph –exactly one of ∅ → { t } and { s } → { s, t } holds. Applying (7) with i = t and j = s , we see thatthis is equivalent to q t ( q t − m ts m ss q s ) = q t − m ts m ss q t q s < . (9)(Here we also use that q is generic, meaning that no expressions in (7) can be 0.) As O is aUSO, ( t, s ) is not an arc in L O ( ∅ ) (recall Figure 7). In the same vein as before, we get that this isequivalent to q s ( q s − m st m tt q t ) = q s − m st m tt q s q t > . (10)Since M is symmetric positive definite, we have that m ss , m tt > m st = m ts , so (9) and(10) imply that 0 < m ss q t < m tt q s . (11)Now consider a path s → t → u in L O ( ∅ ). On top of (11) we then also get0 < m tt q u < m uu q t , and multiplying these inequalities gives m ss q t m tt q u < m tt q s m uu q t ⇔ m ss q u < q s m uu . Iterating this, we get that (11) not only holds when s → t but actually whenever there is a directedpath from s to t . This implies that there cannot be a directed path from a vertex s back to s ,meaning that L O ( ∅ ) is acyclic.Now we are ready to prove the main theorem: every D -cube O has property L. Proof of Theorem 12.
We proceed by induction on n . For n ≤
2, every USO has property L, sowe are done. For n ≥
3, suppose that every ( n − O be an n -dimensional D-cube induced by M and q , where M is symmetric positive definite.The graph L O ( ∅ ) is acyclic by Lemma 13, so it remains to verify that L O ( V ) is acyclic forgiven V (cid:54) = ∅ . Choose k ∈ V . Notice that V is in the ( n − F of O induced by[ { k } , [ n ]], and that L O ( V ) = L F ( V ). Renaming dimensions K = [ n ] \ { j } to [ n −
1] turns F intoan ( n − O (cid:48) , and V into V (cid:48) ⊆ [ n − L O ( V ) is isomorphic to L O (cid:48) ( V (cid:48) ).Orientation O (cid:48) is known to be a P-cube, induced by M (cid:48) K,K (and suitable q (cid:48) K ), where M (cid:48) = M ( { k } ) − M, (12)with M ( { k } ) − as in (8) [29, Property 5]. We also claim that M (cid:48) K,K is symmetric positive definite,which yields the desired result because it implies that O (cid:48) is actually a D-cube of dimension n − L O (cid:48) ( V (cid:48) ) and L O ( V ) are acyclic, as desired.13o show that M (cid:48) K,K is symmetric positive definite, we first observe (simple calculations) that(12) yields M (cid:48) = ( m (cid:48) ij ) with m (cid:48) ij = m ij − m ik m kj m kk , i, j (cid:54) = k. (13)Since M is symmetric, it follows that M (cid:48) K,K is symmetric. Furthermore, for x ∈ R n − , we can easilyverify that x T M (cid:48) K,K x = y T M y ≥
0, where y = ( x , . . . , x k − , − ( (cid:80) i (cid:54) = k m ki x i ) /m kk , x k +1 , . . . , x n ),and thus M (cid:48) K,K is also positive definite.
We have seen that property L is in general not preserved under the application of a cube auto-morphism; see Figure 8. This begs the question: does every USO have an isomorphic copy thatsatisfies property L? If so, property L would in the above sense be satisfied by all USOs, and itwould not really be justified to call the property “combinatorial”.In dimensions 3 and 4, the question can be answered by brute-force, using lists of all isomor-phism classes of USOs. In dimension 3, there are 19 such classes, and each one turns out to containa member with property L. But already in dimension 4, there are 9 (out of 14614) isomorphismclasses that contain no member with property L. Figure 11 depicts a 4-dimensional USO such thatno isomorphic copy has property L.Figure 11: A USO with no isomorphic copy having property L.In this section, we systematically construct such examples in higher dimensions. We show thatfor every n -cube USO, there is a 2 n -cube USO—a kaleidoscope —that contains all “mirror images”of the former. And using this fact, we show that if a n -cube USO does not have property L,then no isomorphic copy of the corresponding kaleidoscope has property L. Separately, startingfrom any P-cube, we show how to construct a kaleidoscope that is also a P-cube, implying thatproperty L is not satisfied by all P-cubes. In particular, starting from the spinner, we construct a6-dimensional P-cube with no isomorphic copy that satisfies property L.Throughout this section, we mostly consider n -cube USOs and 2 n -cube USOs. Whenever weconsider USOs of (some possibly other) general dimension, we will use m to denote this dimension.It will also be convenient to slightly abuse notation and identify a USO with its outmap φ .Two m -cube USOs ψ and ψ (cid:48) are isomorphic if there is a bijection h : 2 [ m ] → [ m ] (a cubeautomorphism) such that for all V, V (cid:48) ⊆ [ m ], we have V → ψ V (cid:48) if and only if h ( V ) → ψ (cid:48) h ( V (cid:48) ). For14xample, the map defined by h ( V ) = V ⊕ F , for some F ⊆ [ m ], is a cube automorphism. It mapsa USO to one of its 2 m mirror images : Definition 14.
Let ψ be an m -cube USO and F ⊆ [ m ] . The USO ψ (cid:48) defined by ψ (cid:48) ( V ) = ψ ( V ⊕ F ) , V ⊆ [ m ] , (14) is the mirror image of ψ along dimensions F . Figure 12 depicts the 8 mirror images of the spinner (one of which is the spinner itself,corresponding to F = ∅ ). Note that mirroring is not to be confused with reversing edges, ψ (cid:48) ( V ) = ψ ( V ) ⊕ R (Definition 3). While the latter operation also preserves the USO property [30,Lemma 2.1], it is in general not an automorphism.Figure 12: The 8 mirror images of the spinner (lower left corner).Actually, all automorphisms can be described as a mirroring automorphism up to a permutationof the dimensions. That is, any m -cube automorphism h is of the form h ( V ) = π ( V ⊕ F ) , V ⊆ [ m ] , where F ⊆ [ m ] is the set of mirrored dimensions, and π : [ m ] → [ m ] is a permutation (renaming)of the dimensions. This follows from the fact that a cube automorphism is determined by theimages of the m + 1 vertices ∅ and { i } for i ∈ [ m ].15ext we define the concept of a kaleidoscope, a USO that connects all mirror images of agiven n -cube USO along n new dimensions. For this, we introduce the following notation. Fora vertex (or outmap value) V ⊆ [2 n ], V L = V ∩ [ n ] denotes the lower dimensions, and V H = { i − n : i ∈ V ∩ { n + 1 , . . . , n }} the upper dimensions, renamed such they also fall into the range[ n ]. For example, if n = 3 and V = { , , , } , then V L = { , } and V H = { , } . Note that( U ⊕ V ) L = U L ⊕ V L and ( U ⊕ V ) H = U H ⊕ V H ; we will use this in arguments below. Definition 15.
Let φ be an n -cube USO and ψ a n -cube USO. The USO ψ is a kaleidoscope for φ if ψ ( V ) L = φ ( V L ⊕ V H ) , ∀ V ⊆ [2 n ] . (15)Figure 13 illustrates a kaleidoscope for the spinner, connecting the mirror images in Figure 12along new dimensions 4 , ,
6. In general, each of the 2 n subcubes with carrier [ n ] of a kaleidoscopeis a particular mirror image of φ , so that ψ contains all possible mirror images of φ . More precisely,if V ⊆ { n + 1 , . . . , n } , then the subcube induced by the interval [ V, V ∪ [ n ]] is the mirror imageof φ along dimensions V H . We will make this containment formal in Definition 17 and Lemma 18below. Figure 13: The kaleidoscope for the spinner resulting from Lemma 16; all edges along dimensions4 , , φ , we can construct a kaleidoscope ψ . For example, if we set ψ ( V ) H = V H for all V , this defines ψ completely, together with (15), and the result is a USO; this is a special16ase of the product construction due to Schurr and Szab´o [25, Lemma 3]. For completeness, weprovide a proof. Lemma 16.
Let φ be a n -cube USO and define ψ : 2 [2 n ] → [2 n ] by ψ ( V ) L := φ ( V L ⊕ V H ) ,ψ ( V ) H := V H . , V ⊆ [2 n ] . Then ψ is a n -cube USO and hence a kaleidoscope for φ .Proof. According to [30, Lemma 2.3], ψ is a USO if and only if ( ψ ( U ) ⊕ ψ ( V )) ∩ ( U ⊕ V ) (cid:54) = ∅ for all U (cid:54) = V . In words, restricted to the subcube spanned by U and V , the two vertices have differentoutmap values. We verify this property for ψ as defined above. Fix U (cid:54) = V . If U H (cid:54) = V H , choose i ∈ U H ⊕ V H = ψ ( U ) H ⊕ ψ ( V ) H = ( ψ ( U ) ⊕ ψ ( V )) H = ( U ⊕ V ) H . Hence, i + n ∈ ( ψ ( U ) ⊕ ψ ( V )) ∩ ( U ⊕ V )).If U H = V H =: W , we have U L (cid:54) = V L and U L ⊕ W =: U (cid:48) (cid:54) = V (cid:48) := V L ⊕ W . Since φ is a USO, thereis i ∈ ( φ ( U (cid:48) ) ⊕ φ ( V (cid:48) )) ∩ ( U (cid:48) ∩ V (cid:48) ), equivalently, i ∈ ( ψ ( U ) L ⊕ ψ ( V ) L ) ∩ ( U L ⊕ V L ) = ( ψ ( U ) ⊕ ψ ( V )) L ∩ ( U ⊕ V ) L ⊆ ( ψ ( U ) ⊕ ψ ( V )) ∩ ( U ⊕ V ) . In both cases, ( ψ ( U ) ⊕ ψ ( V )) ∩ ( U ⊕ V ) (cid:54) = ∅ . Definition 17.
Let φ be an n -cube USO and ψ (cid:48) a n -cube USO. The USO ψ (cid:48) contains φ if thereis V ⊆ { n + 1 , . . . , n } such that ψ (cid:48) ( U ) L = φ ( U L ) , ∀ U ∈ [ V, V ∪ [ n ]] . (16)This means that the subcube of ψ (cid:48) induced by [ V, V ∪ [ n ]] is a “translated copy” of φ . Lemma 18.
Suppose that ψ is a kaleidoscope for the n -cube USO φ . Let F ⊆ [2 n ] , and let ψ (cid:48) bethe mirror image of ψ along dimensions F . Then ψ (cid:48) contains φ .Proof. Let V ⊆ { n + 1 , . . . , n } be the unique set such that V H = F L ⊕ F H . We claim that thesubcube induced by [ V, V ∪ [ n ]] is the one carrying the translated copy of φ . To check this, fix U ∈ [ V, V ∪ [ n ]] and set W = U ⊕ F . Then W L = U L ⊕ F L and W H = U H ⊕ F H . Furthermore, ψ (cid:48) ( U ) L (14) = ψ ( U ⊕ F ) L = ψ ( W ) L (15) = φ ( W L ⊕ W H ) = φ ( U L ⊕ U H ⊕ F L ⊕ F H )= φ ( U L ⊕ U H ⊕ V H ) = φ ( U L ) , using U H = V H for U ∈ [ V, V ∪ [ n ]]. Hence, we have verified (16).Here is the first main result of this section. Theorem 19.
Suppose the n -cube USO ψ is a kaleidoscope for the n -cube USO φ , and furthersuppose that φ fails to have property L. Let ψ (cid:48) be any USO isomorphic to ψ . Then ψ (cid:48) does notsatisfy property L either.Proof. Recall that any m -cube automorphism h is of the form h ( V ) = π ( V ⊕ F ) , V ⊆ [ m ] , (17)17here F ⊆ [ m ] is the set of mirrored dimensions, and π : [ m ] → [ m ] is a permutation (renaming)of dimensions.With m = 2 n , let ψ and ψ (cid:48) be isomorphic under h as in (17). We first consider the case where π = id , the identity permutation. In this case, ψ (cid:48) is the mirror image of ψ along dimensions F ,and by Lemma 18, ψ (cid:48) contains φ in the sense that there is some V ⊆ { n + 1 , . . . , n } such that (16)holds. This shows that the L-graph L ψ (cid:48) ( V ∪ W ) contains L φ ( W ) as a subgraph. Indeed, ( i, j ) isby (1) an arc of L φ ( W ) if and only if j ∈ φ ( W ) ⊕ φ ( W ∪ { i } ) = φ (( V ∪ W ) L ) ⊕ φ (( V ∪ W ∪ { i } ) L ).By (16) applied with U = V ∪ W, V ∪ W ∪ { i } , we then have j ∈ ψ (cid:48) ( V ∪ W ) L ⊕ ψ (cid:48) ( V ∪ W ∪ { i } ) L which for j ∈ [ n ] is equivalent to j ∈ ψ (cid:48) ( V ∪ W ) ⊕ ψ (cid:48) ( V ∪ W ∪ { i } ), so ( i, j ) is also an arc of L ψ (cid:48) ( V ∪ W ). Since L φ ( W ) is cyclic, so is L ψ (cid:48) ( V ∪ W ), and ψ (cid:48) does not have property L.In the general case where π (cid:54) = id , we obtain ψ (cid:48) by first mirroring ψ along the dimensions F (the result does not have property L), and then renaming dimensions according to π . As the latteroperation does not affect property L, ψ (cid:48) does not have property L, either. P-cube kaleidoscopes.
From Theorem 19, we already know that there are USOs without anisomorphic copy that satisfies property L, namely any kaleidoscope of a n -cube USO withoutproperty L. Here we consider the setting in which the n -cube USO is a P-cube. Our goal is tostart with the P-cube and construct a kaleidoscope that is also a P-cube, implying the existenceof P-cubes that have no isomorphic copy with property L.It is not a priori clear how to approach this contruction because it is not known if the kaleido-scopes built from Schurr and Szab´o’s combinatorial product construction [25] are P-cubes. In whatfollows, we develop an algebraic construction that builds a P-cube kaleidoscope from any givenP-cube. We found it somewhat surprising that this works (in a simple way), and the constructionmay be of general interest as a new way to build P-cubes from P-cubes.We start with a construction that “blows up” a P-matrix to twice its dimension. Lemma 20.
Let A be an n × n P-matrix. Then M = (cid:18) A A + IA − I A (cid:19) ∈ R n × n is also a P-matrix.Proof. We use a known characterization of P-matrices [5, Theorem 3.3.4]: M is a P-matrix if andonly if for every nonzero vector z , there exists an index i such that z i ( M z ) i > M does notreverse all signs of z ).So let us consider a nonzero 2 n -vector z = ( x , y ) where x and y are n -vectors. For i ∈ [ n ], wehave z i ( M z ) i = x i ( A x + ( A + I ) y ) i = x i ( A ( x + y )) i + x i y i , (18)and z i + n ( M z ) i + n = y i (( A − I ) x + A y ) i = y i ( A ( x + y )) i − x i y i , (19)If x = − y (cid:54) = , there is an index i such that x i = − y i (cid:54) = 0. In this case, (19) yields z i + n ( M z ) i + n = − x i y i >
0. If x (cid:54) = − y , we add up (18) and (19) to get z i ( M z ) i + z i + n ( M z ) i + n = ( x + y ) i ( A ( x + y )) i . and since A is a P -matrix, there is an index i such that z i ( M z ) i + z i + n ( M z ) i + n >
0. But then,one of z i ( M z ) i and z i + n ( M z ) i + n must be positive as well.18sing the blow-up construction of Lemma 20, along with a suitable right-hand side q , we cannow construct a P-cube kaleidoscope from any P-cube. Theorem 21.
Let A ∈ R n × n be a P -matrix, b ∈ R n generic. Let φ be the n -dimensional P-cubedefined by A and b ; see Definition 10. Let M = (cid:18) A A + IA − I A (cid:19) ∈ R n × n and q = (cid:18) bb (cid:19) ∈ R n . Then M and q define a n -dimensional P-cube ψ . Furthermore, ψ is a kaleidoscope for φ .Proof. We already know from Lemma 20 that M is a P-matrix. If q is generic, M and q definea P-cube ψ . For the kaleidoscope property, we will compute the outmap values ψ ( V ) , V ⊆ [2 n ].In doing so, we will also see that q is indeed generic. Let us fix V for the remainder of the proof.We recall from Section 4 that ψ ( V ) is determined by the signs of the variables not prescribed by V . More concretely, there are unique w and z that solve the system of equations w − M z = q ; w i = 0 , i ∈ V ; z i = 0 , i / ∈ V . And the outmap value of V is ψ ( V ) = { i : z i < w i < } , (20)where q being generic means that z i = w i = 0 can never happen. Hence, for M and q as in thestatement of the theorem, there are unique w , w (cid:48) , z , z (cid:48) ∈ R n such that (cid:18) ww (cid:48) (cid:19) − (cid:18) A A + IA − I A (cid:19) (cid:18) zz (cid:48) (cid:19) = (cid:18) bb (cid:19) , (21) w i = 0 , i ∈ V L , (22) w (cid:48) i = 0 , i ∈ V H , (23) z i = 0 , i / ∈ V L , (24) z (cid:48) i = 0 , i / ∈ V H . (25)The outmap (20) is then determined by ψ ( V ) L = { i ∈ [ n ] : z i < w i < } , (26) ψ ( V ) H = { i ∈ [ n ] : z (cid:48) i < w (cid:48) i < } . (27)Expanding (21), we get w − z (cid:48) − A ( z + z (cid:48) ) = b , (28) w (cid:48) + z − A ( z + z (cid:48) ) = b , (29)and it follows that w − z (cid:48) = w (cid:48) + z := y , (30) z + z (cid:48) = w − w (cid:48) := x . (31)With this and (22) through (25), we can summarize the situation as in Figure 14.19 n] z i = z (cid:48) i = 0 x i = 0 , w i = y i ⇒ V L V H w i = w (cid:48) i = 0 x i = 0 ,w i = z (cid:48) i = 0 w (cid:48) i = z i = 0 y i = 0 , y i = 0 , ⇒ ⇒ ⇒ z i = x i z i = y i w i = x i Figure 14: Proof of Theorem 21: Prescribed variables and implied equalitiesFrom (30), (31), and Figure 14, we see that x and y (uniquely) solve the system y − A x = b ; y i = 0 , i ∈ V L ⊕ V H ; x i = 0 , i / ∈ V L ⊕ V H . Hence, the P-cube φ determined by A and b bydefinition satisfies φ ( V L ⊕ V H ) = { i ∈ [ n ] : x i < y i < } . (32)To conclude that ψ is a kaleidoscope for φ , it remains to show that q is generic, and that thekaleidoscope property (15) holds, meaning the outmap value in (32) equals the one in (26). Forthe latter, we have to prove that x i < y i < ⇔ z i < w i < . (33)This easily follows from the equalities depicted in Figure 14. If x i <
0, we have z i = x i < w i = x i <
0. If y i <
0, we have z i = y i < w i = y i <
0. Vice versa, if z i <
0, then x i = z i < y i = z i <
0; and if w i <
0, then x i = w i < y i = w i < q is generic, we need to show that z i = w i = 0 and z (cid:48) i = w (cid:48) i = 0 can never happen.Adding up (30) and (31) yields w + z = x + y . Since b is generic, we know that x i + y i (cid:54) = 0 for all i (one of the values is 0, the other one isn’t). Hence, also w i + z i (cid:54) = 0 for all i . Again, one of thesetwo values is 0, so w i − z i (cid:54) = 0 for all i as well, and with (30), z (cid:48) i + w (cid:48) i (cid:54) = 0 follows for all i .As an example, the kaleidoscope for the spinner, depicted in Figure 13, is generated by M = , q = , following the construction of Theorem 21 for the matrix A and vector b that generate the spinneraccording to (6). The main research question this work raises is whether we can algorithmically make use of prop-erty L. Ideally, we would like to exploit the property towards speeding up sink-finding for D-cubes.20n the one hand, it seems discouraging that property L is in general not invariant under ap-plying cube automorphisms. In the “abstract USO world”, pairs of isomorphic USOs are typicallyconsidered “the same”, and many algorithms in fact perform on them in the same way. Suchalgorithms do not have much potential to exploit the input’s property L when another isomorphiccopy fails to satisfy it.On the other hand, the insight here may be that algorithms tailored towards D-cubes shouldhave a “sense of direction” and not ignore how vertices are labeled.There is a family of USO algorithms commonly summarized under the term product algo-rithm [30, Lemma 3.2]. In order to be able to apply them to a class U of USOs, two propertiesof U are required: U must be closed under taking subcubes, but also under taking inherited ori-entations [30, Section 3]. The class of all USOs meets both requirements. The class of D-cubesis closed under taking subcubes (this implicitly follows from the proof of Theorem 12); but it isunknown whether the class is closed under taking inherited orientations (probably, it’s not).Finding weaker closure properties and corresponding algorithms with a “sense of direction”might be a way to make progress here. References [1] M. L. Balinski. On the graph structure of convex polyhedra in n -space. Pacific J. Math. ,11(2):431–434, 1961.[2] Vitor Bosshard and Bernd G¨artner. Pseudo unique sink orientations. https://arxiv.org/abs/1704.08481 , 2017.[3] Ramaswamy Chandrasekaran. A special case of the complementary pivot problem.
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