Computer Science

We prove a general structural theorem for a wide family of local algorithms, which includes property testers, local decoders, and PCPs of proximity. Namely, we show that the structure of every algorithm that makes q adaptive queries and satisfies a natural robustness condition admits a sample-based algorithm with n 1−1/O( q 2 log 2 q) sample complexity, following the definition of Goldreich and Ron (TOCT 2016). We prove that this transformation is nearly optimal. Our theorem also admits a scheme for constructing privacy-preserving local algorithms. Using the unified view that our structural theorem provides, we obtain results regarding various types of local algorithms, including the following. - We strengthen the state-of-the-art lower bound for relaxed locally decodable codes, obtaining an exponential improvement on the dependency in query complexity; this resolves an open problem raised by Gur and Lachish (SODA 2020). - We show that any (constant-query) testable property admits a sample-based tester with sublinear sample complexity; this resolves a problem left open in a work of Fischer, Lachish, and Vasudev (FOCS 2015) by extending their main result to adaptive testers. - We prove that the known separation between proofs of proximity and testers is essentially maximal; this resolves a problem left open by Gur and Rothblum (ECCC 2013, Computational Complexity 2018) regarding sublinear-time delegation of computation. Our techniques strongly rely on relaxed sunflower lemmas and the Hajnal-Szemerédi theorem.

We prove two new results about the randomized query complexity of composed functions. First, we show that the randomized composition conjecture is false: there are families of partial Boolean functions f and g such that R(f∘g)≪R(f)R(g) . In fact, we show that the left hand side can be polynomially smaller than the right hand side (though in our construction, both sides are polylogarithmic in the input size of f ). Second, we show that for all f and g , R(f∘g)=Ω(noisyR(f)⋅R(g)) , where noisyR(f) is a measure describing the cost of computing f on noisy oracle inputs. We show that this composition theorem is the strongest possible of its type: for any measure M(⋅) satisfying R(f∘g)=Ω(M(f)R(g)) for all f and g , it must hold that noisyR(f)=Ω(M(f)) for all f . We also give a clean characterization of the measure noisyR(f) : it satisfies noisyR(f)=Θ(R(f∘gapma j n )/R(gapma j n )) , where n is the input size of f and gapma j n is the n − − √ -gap majority function on n bits.

The classes PPA- p have attracted attention lately, because they are the main candidates for capturing the complexity of Necklace Splitting with p thieves, for prime p . However, these classes were not known to have complete problems of a topological nature, which impedes any progress 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, Chen et al., 2009b] and the PPA-completeness of Necklace Splitting with 2 thieves [Filos-Ratsikas and Goldberg, 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's Lemma, termed p -polygon-Tucker, as well as the associated Borsuk-Ulam-type theorem, p -polygon-Borsuk-Ulam, are PPA- p -complete. Then, we show that the computational version of the well-known BSS Theorem [Barany et al., 1981], as well as the associated BSS-Tucker problem are PPA- p -complete. Finally, using a different generalization of Tucker's Lemma (termed Z p -star-Tucker), which we prove to be PPA- p -complete, we prove that p -thief Necklace Splitting is in PPA- p . This latter result gives a new combinatorial proof for the Necklace 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 proof of Tucker's lemma by Freund and Todd [1981]. We believe that this new proof technique is of independent interest.

Given a graph G and a digraph D whose vertices are the edges of G , we investigate the problem of finding a spanning tree of G that satisfies the constraints imposed by D . The restrictions to add an edge in the tree depend on its neighborhood in D . Here, we generalize previously investigated problems by also considering as input functions ℓ and u on E(G) that give a lower and an upper bound, respectively, on the number of constraints that must be satisfied by each edge. The produced feasibility problem is denoted by \texttt{G-DCST}, while the optimization problem is denoted by \texttt{G-DCMST}. We show that \texttt{G-DCST} is NP-complete even under strong assumptions on the structures of G and D , as well as on functions ℓ and u . On the positive side, we prove two polynomial results, one for \texttt{G-DCST} and another for \texttt{G-DCMST}, and also give a simple exponential-time algorithm along with a proof that it is asymptotically optimal under the Ð. Finally, we prove that other previously studied constrained spanning tree (\textsc{CST}) problems can be modeled within our framework, namely, the \textsc{Conflict CST}, the \textsc{Forcing CS, the \textsc{At Least One/All Dependency CST}, the \textsc{Maximum Degree CST}, the \textsc{Minimum Degree CST}, and the \textsc{Fixed-Leaves Minimum Degree CST}.

The CMI Millennium "P vs NP Problem" can be resolved e.g. if one shows at least one counterexample to the conjecture "P is equal to NP". A certain class of problems being such counterexamples is formulated. This implies the rejection of the hypothesis "P is equal to NP" for any conditions satisfying the formulation of the problem. Thus, the solution "P is different from NP" of the problem is proved. The class of counterexamples can be interpreted as any quantum superposition of any finite set of quantum states. The Kochen-Specker theorem is involved. Any fundamentally random choice among a finite set of alternatives belong to NP, but not to P. The conjecture that the set complement of P to NP can be described by that kind of choice is formulated exhaustively.

We consider the complexity of counting weighted graph homomorphisms defined by a symmetric matrix A . Each symmetric matrix A defines a graph homomorphism function Z A (⋅) , also known as the partition function. Dyer and Greenhill [10] established a complexity dichotomy of Z A (⋅) for symmetric {0,1} -matrices A , and they further proved that its #P-hardness part also holds for bounded degree graphs. Bulatov and Grohe [4] extended the Dyer-Greenhill dichotomy to nonnegative symmetric matrices A . However, their hardness proof requires graphs of arbitrarily large degree, and whether the bounded degree part of the Dyer-Greenhill dichotomy can be extended has been an open problem for 15 years. We resolve this open problem and prove that for nonnegative symmetric A , either Z A (G) is in polynomial time for all graphs G , or it is #P-hard for bounded degree (and simple) graphs G . We further extend the complexity dichotomy to include nonnegative vertex weights. Additionally, we prove that the #P-hardness part of the dichotomy by Goldberg et al. [12] for Z A (⋅) also holds for simple graphs, where A is any real symmetric matrix.

In a non-uniform Constraint Satisfaction problem CSP(G), where G is a set of relations on a finite set A, the goal is to find an assignment of values to variables subject to constraints imposed on specified sets of variables using the relations from G. The Dichotomy Conjecture for the non-uniform CSP states that for every constraint language G the problem CSP(G) is either solvable in polynomial time or is NP-complete. It was proposed by Feder and Vardi in their seminal 1993 paper. In this paper we confirm the Dichotomy Conjecture.

In cellular automata with multiple speeds for each cell i there is a positive integer p i such that this cell updates its state still periodically but only at times which are a multiple of p i . Additionally there is a finite upper bound on all p i . Manzoni and Umeo have described an algorithm for these (one-dimensional) cellular automata which solves the Firing Squad Synchronization Problem. This algorithm needs linear time (in the number of cells to be synchronized) but for many problem instances it is slower than the optimum time by some positive constant factor. In the present paper we derive lower bounds on possible synchronization times and describe an algorithm which is never slower and in some cases faster than the one by Manzoni and Umeo and which is close to a lower bound (up to a constant summand) in more cases.

Given an integer n?? and an irreducible character ? λ of S n for some partition λ of n , the immanant imm λ : C n?n ?�C maps matrices A??C n?n to imm λ (A)= ?????S n ? λ (?) ??n i=1 A i,?(i) . Important special cases include the determinant and permanent, which are the immanants associated with the sign and trivial character, respectively. It is known that immanants can be evaluated in polynomial time for characters that are close to the sign character: Given a partition λ of n with s parts, let b(λ):=n?�s count the boxes to the right of the first column in the Young diagram of λ . For a family of partitions ? , let b(?):= max λ?��?b(λ) and write Imm (?) for the problem of evaluating imm λ (A) on input A and λ?��?. If b(?)<??, then Imm (?) is known to be polynomial-time computable. This subsumes the case of the determinant. On the other hand, if b(?)=??, then previously known hardness results suggest that Imm (?) cannot be solved in polynomial time. However, these results only address certain restricted classes of families ? . In this paper, we show that the parameterized complexity assumption FPT ??#W[1] rules out polynomial-time algorithms for Imm (?) for any computationally reasonable family of partitions ? with b(?)=??. We give an analogous result in algebraic complexity under the assumption VFPT ??VW[1]. Furthermore, if b(λ) even grows polynomially in ? , we show that Imm (?) is hard for #P and VNP. This concludes a series of partial results on the complexity of immanants obtained over the last 35 years.

In this work we prove a version of the Sylvester-Gallai theorem for quadratic polynomials that takes us one step closer to obtaining a deterministic polynomial time algorithm for testing zeroness of Σ [3] ΠΣ Π [2] circuits. Specifically, we prove that if a finite set of irreducible quadratic polynomials Q satisfy that for every two polynomials Q 1 , Q 2 ∈Q there is a subset K⊂Q , such that Q 1 , Q 2 ∉K and whenever Q 1 and Q 2 vanish then also ∏ i∈K Q i vanishes, then the linear span of the polynomials in Q has dimension O(1) . This extends the earlier result [Shpilka19] that showed a similar conclusion when |K|=1 . An important technical step in our proof is a theorem classifying all the possible cases in which a product of quadratic polynomials can vanish when two other quadratic polynomials vanish. I.e., when the product is in the radical of the ideal generates by the two quadratics. This step extends a result from [Shpilka19]that studied the case when one quadratic polynomial is in the radical of two other quadratics.