Proceedings of the 36th Computational Complexity Conference最新文献

{"title":"Hitting sets and reconstruction for dense orbits in VPe and ΣΠΣ circuits","authors":"D. Medini, Amir Shpilka","doi":"10.4230/LIPIcs.CCC.2021.19","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.19","url":null,"abstract":"In this paper we study polynomials in VPe (polynomial-sized formulas) and in ΣΠΣ (polynomial-size depth-3 circuits) whose orbits, under the action of the affine group GLaffn (F) (the action of (A, b) ∈ GLaffn (F) on a polynomial f ∈ F[x] is defined as (A, b) ο f = f (AT x + b)), are dense in their ambient class. We construct hitting sets and interpolating sets for these orbits as well as give reconstruction algorithms. Specifically, we obtain the following results: 1. For [EQUATION], where the ℓis are linearly independent linear functions, we construct a polynomial-sized interpolating set, and give a polynomial-time reconstruction algorithm. By a result of Bringmann, Ikenmeyer and Zuiddam, the set of all such polynomials is dense in VPe [14], thus our construction gives the first polynomial-size interpolating set for a dense subclass of VPe. 2. For polynomials of the form ANFΔ (ℓ1(x),..., ℓ4Δ(x)) where ANFΔ(x) is the canonical read-once formula in alternating normal form, of depth 2Δ, and the ℓis are linearly independent linear functions, we provide a quasipolynomial-size interpolating set. We also observe that the reconstruction algorithm of [35] works for all polynomials in this class. This class is also dense in VPe. 3. Similarly, we give a quasipolynomial-sized hitting set for read-once formulas (not necessarily in alternating normal form) composed with a set of linearly independent linear functions. This gives another dense class in VPe. 4. We give a quasipolynomial-sized hitting set for polynomials of the form f(ℓ1(x),..., ℓm(x)), where f is an m-variate s-sparse polynomial. and the ℓis are linearly independent linear functions in n ≥ m variables. This class is dense in ΣΠΣ. 5. For polynomials of the form Σsi=1 Πdj=1 ℓi,j (x), where the ℓi,js are linearly independent linear functions, we construct a polynomial-sized interpolating set. We also observe that the reconstruction algorithm of [45] works for every polynomial in the class. This class is dense in ΣΠΣ. As VP = VNC2, our results for VPe translate immediately to VP with a quasipolynomial blow up in parameters. If any of our hitting or interpolating sets could be made robust then this would immediately yield a hitting set for the superclass in which the relevant class is dense, and as a consequence also a lower bound for the superclass. Unfortunately, we also prove that the kind of constructions that we have found (which are defined in terms of k-independent polynomial maps) do not necessarily yield robust hitting sets.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123773539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Error reduction for weighted PRGs against read once branching programs","authors":"Gil Cohen, Dean Doron, Oren Renard, Ori Sberlo, A. Ta-Shma","doi":"10.4230/LIPIcs.CCC.2021.22","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.22","url":null,"abstract":"Weighted pseudorandom generators (WPRGs), introduced by Braverman, Cohen and Garg [5], are a generalization of pseudorandom generators (PRGs) in which arbitrary real weights are considered, rather than a probability mass. Braverman et al. constructed WPRGs against read once branching programs (ROBPs) with near-optimal dependence on the error parameter. Chattopadhyay and Liao [6] somewhat simplified the technically involved BCG construction, also obtaining some improvement in parameters. In this work we devise an error reduction procedure for PRGs against ROBPs. More precisely, our procedure transforms any PRG against length n width w ROBP with error 1/poly(n) having seed length s to a WPRG with seed length s + O(log w/ε · log log 1/ε). By instantiating our procedure with Nisan's PRG [17] we obtain a WPRG with seed length O(log n · log(nw) + log w/ε · log log 1/ε). This improves upon [5] and is incomparable with [6]. Our construction is significantly simpler on the technical side and is conceptually cleaner. Another advantage of our construction is its low space complexity O(log nw) + poly(log log 1/ε) which is logarithmic in n for interesting values of the error parameter ε. Previous constructions (like [5, 6]) specify the seed length but not the space complexity, though it is plausible they can also achieve such (or close) space complexity.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121711358","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Branching programs with bounded repetitions and flow formulas","authors":"A. Sofronova, Dmitry Sokolov","doi":"10.4230/LIPIcs.CCC.2021.17","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.17","url":null,"abstract":"Restricted branching programs capture various complexity measures like space in Turing machines or length of proofs in proof systems. In this paper, we focus on the application in the proof complexity that was discovered by Lovasz et al. [14] who showed the equivalence between regular Resolution and read-once branching programs for \"unsatisfied clause search problem\" (Searchφ). This connection is widely used, in particular, in the recent breakthrough result about the Clique problem in regular Resolution by Atserias et al. [5]. We study the branching programs with bounded repetitions, so-called (1, +k)-BPs (Sieling [21]) in application to the Searchφ problem. On the one hand, it is a natural generalization of read-once branching programs. On the other hand, this model gives a powerful proof system that can efficiently certify the unsatisfiability of a wide class of formulas that is hard for Resolution (Knop [13]). We deal with Searchφ that is \"relatively easy\" compared to all known hard examples for the (1, +k)-BPs. We introduce the first technique for proving exponential lower bounds for the (1, +k)-BPs on Searchφ. To do it we combine a well-known technique for proving lower bounds on the size of branching programs [12,21,22] with the modification of the \"closure\" technique [1,3]. In contrast with most Resolution lower bounds, our technique uses not only \"local\" properties of the formula, but also a \"global\" structure. Our hard examples are based on the Flow formulas introduced in [3].","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130734858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Hardness of constant-round communication complexity","authors":"Shuichi Hirahara, Rahul Ilango, B. Loff","doi":"10.4230/LIPIcs.CCC.2021.31","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.31","url":null,"abstract":"How difficult is it to compute the communication complexity of a two-argument total Boolean function f : [N] X [N] → {0, 1}, when it is given as an N X N binary matrix? In 2009, Kushilevitz and Weinreb showed that this problem is cryptographically hard, but it is still open whether it is NP-hard. In this work, we show that it is NP-hard to approximate the size (number of leaves) of the smallest constant-round protocol for a two-argument total Boolean function f : [N] X [N] → {0, 1}, when it is given as an N X N binary matrix. Along the way to proving this, we show a new deterministic variant of the round elimination lemma, which may be of independent interest.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123421298","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"An improved protocol for the exactly-N problem*","authors":"N. Linial, A. Shraibman","doi":"10.4230/LIPIcs.CCC.2021.2","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.2","url":null,"abstract":"In the 3-players exactly-N problem the players need to decide whether x + y + z = N for inputs x, y, z and fixed N. This is the first problem considered in the multiplayer Number On the Forehead (NOF) model. Even though this is such a basic problem, no progress has been made on it throughout the years. Only recently have explicit protocols been found for the first time, yet no improvement in complexity has been achieved to date. The present paper offers the first improved protocol for the exactly-N problem. This improved protocol has also interesting consequences in additive combinatorics. As we explain below, it yields a higher lower bound on the possible density of corner-free sets in [N] × [N].","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129460729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"A majority lemma for randomised query complexity","authors":"Mika Göös, Gilbert Maystre","doi":"10.4230/LIPIcs.CCC.2021.18","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.18","url":null,"abstract":"We show that computing the majority of n copies of a boolean function g has randomised query complexity [EQUATION]. In fact, we show that to obtain a similar result for any composed function f ο gn, it suffices to prove a sufficiently strong form of the result only in the special case g = GAPOr.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132985217","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The power of negative reasoning","authors":"Susanna F. de Rezende, M. Lauria, J. Nordström, Dmitry Sokolov","doi":"10.4230/LIPIcs.CCC.2021.40","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.40","url":null,"abstract":"Semialgebraic proof systems have been studied extensively in proof complexity since the late 1990s to understand the power of Gröbner basis computations, linear and semidefinite programming hierarchies, and other methods. Such proof systems are defined alternately with only the original variables of the problem and with special formal variables for positive and negative literals, but there seems to have been no study how these different definitions affect the power of the proof systems. We show for Nullstellensatz, polynomial calculus, Sherali-Adams, and sums-of-squares that adding formal variables for negative literals makes the proof systems exponentially stronger, with respect to the number of terms in the proofs. These separations are witnessed by CNF formulas that are easy for resolution, which establishes that polynomial calculus, Sherali-Adams, and sums-of-squares cannot efficiently simulate resolution without having access to variables for negative literals.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134574788","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Matrix rigidity depends on the target field","authors":"L. Babai, Bohdan Kivva","doi":"10.4230/LIPIcs.CCC.2021.41","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.41","url":null,"abstract":"The rigidity of a matrix A for target rank r is the minimum number of entries of A that need to be changed in order to obtain a matrix of rank at most r (Valiant, 1977). We study the dependence of rigidity on the target field. We consider especially two natural regimes: when one is allowed to make changes only from the field of definition of the matrix (\"strict rigidity\"), and when the changes are allowed to be in an arbitrary extension field (\"absolute rigidity\"). We demonstrate, apparently for the first time, a separation between these two concepts. We establish a gap of a factor of 3/2 − o(1) between strict and absolute rigidities. The question seems especially timely because of recent results by Dvir and Liu (Theory of Computing, 2020) where important families of matrices, previously expected to be rigid, are shown not to be absolutely rigid, while their strict rigidity remains open. Our lower-bound method combines elementary arguments from algebraic geometry with \"untouched minors\" arguments. Finally, we point out that more families of long-time rigidity candidates fall as a consequence of the results of Dvir and Liu. These include the incidence matrices of projective planes over finite fields, proposed by Valiant as candidates for rigidity over F2.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131760665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Proof complexity of natural formulas via communication arguments","authors":"D. Itsykson, Artur Riazanov","doi":"10.4230/LIPIcs.CCC.2021.3","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.3","url":null,"abstract":"A canonical communication problem Search (φ) is defined for every unsatisfiable CNF φ: an assignment to the variables of φ is partitioned among the communicating parties, they are to find a clause of φ falsified by this assignment. Lower bounds on the randomized k-party communication complexity of Search (φ) in the number-on-forehead (NOF) model imply tree-size lower bounds, rank lower bounds, and size-space tradeoffs for the formula φ in the semantic proof system Tcc (k, c) that operates with proof lines that can be computed by k-party randomized communication protocol using at most c bits of communication [9]. All known lower bounds on Search (φ) (e.g. [1, 9, 13]) are realized on ad-hoc formulas φ (i.e. they were introduced specifically for these lower bounds). We introduce a new communication complexity approach that allows establishing proof complexity lower bounds for natural formulas. First, we demonstrate our approach for two-party communication and apply it to the proof system Res(⊕) that operates with disjunctions of linear equalities over F2 [14]. Let a formula PMG encode that a graph G has a perfect matching. If G has an odd number of vertices, then PMG has a tree-like Res(⊕)-refutation of a polynomial-size [14]. It was unknown whether this is the case for graphs with an even number of vertices. Using our approach we resolve this question and show a lower bound 2Ω(n) on size of tree-like Res(⊕)-refutations of PMKn+2,n. Then we apply our approach for k-party communication complexity in the NOF model and obtain a [EQUATION] lower bound on the randomized k-party communication complexity of Search (BPHPM2n) w.r.t. to some natural partition of the variables, where BPHPM2n is the bit pigeonhole principle and M = 2n + 2n(1--1/k). In particular, our result implies that the bit pigeonhole requires exponential tree-like Th(k) proofs, where Th(k) is the semantic proof system operating with polynomial inequalities of degree at most k and k = O(log1--ϵ n) for some ϵ > 0. We also show that BPHP2n+12n superpolynomially separates tree-like Th(log1--ϵ m) from tree-like Th(log m), where m is the number of variables in the refuted formula.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130108532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal tiling of the euclidean space using permutation-symmetric bodies","authors":"M. Braverman, Dor Minzer","doi":"10.4230/LIPIcs.CCC.2021.5","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.5","url":null,"abstract":"What is the least surface area of a permutation-symmetric body B whose Zn translations tile Rn? Since any such body must have volume 1, the isoperimetric inequality implies that its surface area must be at least [EQUATION]. Remarkably, Kindler et al. showed that for general bodies B this is tight, i.e. that there is a tiling body of Rn whose surface area is [EQUATION]. In theoretical computer science, the tiling problem is intimately related to the study of parallel repetition theorems (which are an important component in PCPs), and more specifically in the question of whether a \"strong version\" of the parallel repetition theorem holds. Raz showed, using the odd cycle game, that strong parallel repetition fails in general, and subsequently these ideas were used in order to construct non-trivial tilings of Rn. In this paper, motivated by the study of a symmetric parallel repetition, we consider the permutation-symmetric variant of the tiling problem in Rn. We show that any permutation-symmetric body that tiles Rn must have surface area at least [EQUATION], and that this bound is tight, i.e. that there is a permutation-symmetric tiling body of Rn with surface area [EQUATION]. We also give matching bounds for the value of the symmetric parallel repetition of Raz's odd cycle game. Our result suggests that while strong parallel repetition fails in general, there may be important special cases where it still applies.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128606477","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}