Proceedings of the 36th Computational Complexity Conference最新文献

{"title":"Robustly self-ordered graphs: constructions and applications to property testing","authors":"Oded Goldreich, A. Wigderson","doi":"10.4230/LIPIcs.CCC.2021.12","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.12","url":null,"abstract":"A graph G is called self-ordered (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from G to any graph that is isomorphic to G. We say that G = (V, E) is robustly self-ordered if the size of the symmetric difference between E and the edge-set of the graph obtained by permuting V using any permutation π : V → V is proportional to the number of non-fixed-points of π. In this work, we initiate the study of the structure, construction and utility of robustly self-ordered graphs. We show that robustly self-ordered bounded-degree graphs exist (in abundance), and that they can be constructed efficiently, in a strong sense. Specifically, given the index of a vertex in such a graph, it is possible to find all its neighbors in polynomial-time (i.e., in time that is poly-logarithmic in the size of the graph). We provide two very different constructions, in tools and structure. The first, a direct construction, is based on proving a sufficient condition for robust self-ordering, which requires that an auxiliary graph is expanding. The second construction is iterative, boosting the property of robust self-ordering from smaller to larger graphs. Structuraly, the first construction always yields expanding graphs, while the second construction may produce graphs that have many tiny (sub-logarithmic) connected components. We also consider graphs of unbounded degree, seeking correspondingly unbounded robustness parameters. We again demonstrate that such graphs (of linear degree) exist (in abundance), and that they can be constructed efficiently, in a strong sense. This turns out to require very different tools. Specifically, we show that the construction of such graphs reduces to the construction of non-malleable two-source extractors (with very weak parameters but with some additional natural features). We demonstrate that robustly self-ordered bounded-degree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the bounded-degree and the dense graph models. Indeed, their robustness offers efficient, local and distance preserving reductions from testing problems on ordered structures (like sequences) to the unordered (effectively unlabeled) graphs. One of the results that we obtain, via such a reduction, is a subexponential separation between the query complexities of testing and tolerant testing of graph properties in the bounded-degree graph model.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"19 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":"114589187","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":"On p-group isomorphism: search-to-decision, counting-to-decision, and nilpotency class reductions via tensors","authors":"Joshua A. Grochow, Youming Qiao","doi":"10.4230/LIPIcs.CCC.2021.16","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.16","url":null,"abstract":"In this paper we study some classical complexity-theoretic questions regarding GROUP ISOMORPHISM (GPI). We focus on p-groups (groups of prime power order) with odd p, which are believed to be a bottleneck case for GPI, and work in the model of matrix groups over finite fields. Our main results are as follows. • Although search-to-decision and counting-to-decision reductions have been known for over four decades for GRAPH ISOMORPHISM (GI), they had remained open for GPI, explicitly asked by Arvind & Torán (Bull. EATCS, 2005). Extending methods from TENSOR ISOMORPHISM (Grochow & Qiao, ITCS 2021), we show moderately exponential-time such reductions within p-groups of class 2 and exponent p. • Despite the widely held belief that p-groups of class 2 and exponent p are the hardest cases of GPI, there was no reduction to these groups from any larger class of groups. Again using methods from TENSOR ISOMORPHISM (ibid.), we show the first such reduction, namely from isomorphism testing of p-groups of \"small\" class and exponent p to those of class two and exponent p. For the first results, our main innovation is to develop linear-algebraic analogues of classical graph coloring gadgets, a key technique in studying the structural complexity of GI. Unlike the graph coloring gadgets, which support restricting to various subgroups of the symmetric group, the problems we study require restricting to various subgroups of the general linear group, which entails significantly different and more complicated gadgets. The analysis of one of our gadgets relies on a classical result from group theory regarding random generation of classical groups (Kantor & Lubotzky, Geom. Dedicata, 1990). For the nilpotency class reduction, we combine a runtime analysis of the Lazard Correspondence with TENSOR ISOMORPHISM-completeness results (Grochow & Qiao, ibid.).","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"263 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":"133916694","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":"Rate amplification and query-efficient distance amplification for linear LCC and LDC","authors":"Gil Cohen, Tal Yankovitz","doi":"10.4230/LIPIcs.CCC.2021.1","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.1","url":null,"abstract":"The main contribution of this work is a rate amplification procedure for LCC. Our procedure converts any q-query linear LCC, having rate ρ and, say, constant distance to an asymptotically good LCC with qpoly(1/ρ) queries. Our second contribution is a distance amplification procedure for LDC that converts any linear LDC with distance δ and, say, constant rate to an asymptotically good LDC. The query complexity only suffers a multiplicative overhead that is roughly equal to the query complexity of a length 1/δ asymptotically good LDC. This improves upon the poly(1/δ) overhead obtained by the AEL distance amplification procedure [2, 1]. Our work establishes that the construction of asymptotically good LDC and LCC is reduced, with a minor overhead in query complexity, to the problem of constructing a vanishing rate linear LCC and a (rapidly) vanishing distance linear LDC, respectively.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"50 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":"122673984","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":"Arithmetic circuit complexity of division and truncation","authors":"P. Dutta, Gorav Jindal, Anurag Pandey, Amit Sinhababu","doi":"10.4230/LIPIcs.CCC.2021.25","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.25","url":null,"abstract":"Given polynomials f, g, h ∈ F[x1, ..., xn] such that f = g/h, where both g and h are computable by arithmetic circuits of size s, we show that f can be computed by a circuit of size poly(s, deg(h)). This solves a special case of division elimination for high-degree circuits (Kaltofen'87 & WACT'16). The result is an exponential improvement over Strassen's classic result (Strassen'73) when deg(h) is poly(s) and deg(f) is exp(s), since the latter gives an upper bound of poly(s, deg(f)). Further, we show that any univariate polynomial family (fd)d, defined by the initial segment of the power series expansion of rational function gd(x)/hd(x) up to degree d (i.e. fd = gd/hd mod xd+1), where circuit size of g is sd and degree of gd is at most d, can be computed by a circuit of size poly(sd, deg(hd), log d). We also show a hardness result when the degrees of the rational functions are high (i.e. Ω(d)), assuming hardness of the integer factorization problem. Finally, we extend this conditional hardness to simple algebraic functions as well, and show that for every prime p, there is an integral algebraic power series with its minimal polynomial satisfying a degree p polynomial equation, such that its initial segment is hard to compute unless integer factoring is easy, or a multiple of n! is easy to compute. Both, integer factoring and computation of multiple of n!, are believed to be notoriously hard. In contrast, we show examples of transcendental power series whose initial segments are easy to compute.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"81 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":"116879426","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":"Pseudodistributions that beat all pseudorandom generators (extended abstract)","authors":"Edward Pyne, S. Vadhan","doi":"10.4230/LIPIcs.CCC.2021.33","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.33","url":null,"abstract":"A recent paper of Braverman, Cohen, and Garg (STOC 2018) introduced the concept of a weighted pseudorandom generator (WPRG), which amounts to a pseudorandom generator (PRG) whose outputs are accompanied with real coefficients that scale the acceptance probabilities of any potential distinguisher. They gave an explicit construction of WPRGs for ordered branching programs whose seed length has a better dependence on the error parameter ε than the classic PRG construction of Nisan (STOC 1990 and Combinatorica 1992). In this work, we give an explicit construction of WPRGs that achieve parameters that are impossible to achieve by a PRG. In particular, we construct a WPRG for ordered permutation branching programs of unbounded width with a single accept state that has seed length Õ(log3/2 n) for error parameter ε = 1/poly(n), where n is the input length. In contrast, recent work of Hoza et al. (ITCS 2021) shows that any PRG for this model requires seed length Ω(log2 n) to achieve error ε = 1/poly(n). As a corollary, we obtain explicit WPRGs with seed length Õ(log3/2 n) and error ε = 1/poly(n) for ordered permutation branching programs of width w = poly(n) with an arbitrary number of accept states. Previously, seed length o(log2 n) was only known when both the width and the reciprocal of the error are subpolynomial, i.e. w = no(1) and ε = 1/no(1) (Braverman, Rao, Raz, Yehudayoff, FOCS 2010 and SICOMP 2014). The starting point for our results are the recent space-efficient algorithms for estimating random-walk probabilities in directed graphs by Ahmadenijad, Kelner, Murtagh, Peebles, Sidford, and Vadhan (FOCS 2020), which are based on spectral graph theory and space-efficient Laplacian solvers. We interpret these algorithms as giving WPRGs with large seed length, which we then derandomize to obtain our results. We also note that this approach gives a simpler proof of the original result of Braverman, Cohen, and Garg, as independently discovered by Cohen, Doron, Renard, Sberlo, and Ta-Shma (these proceedings).","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"74 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":"117114718","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":"Deterministic identity testing paradigms for bounded top-fanin depth-4 circuits","authors":"P. Dutta, Prateek Dwivedi, Nitin Saxena","doi":"10.4230/LIPIcs.CCC.2021.11","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.11","url":null,"abstract":"Polynomial Identity Testing (PIT) is a fundamental computational problem. The famous depth-4 reduction (Agrawal & Vinay, FOCS'08) has made PIT for depth-4 circuits, an enticing pursuit. The largely open special-cases of sum-product-of-sum-of-univariates (Σ[k]ΠΣ∧) and sum-product-of-constant-degree-polynomials (Σ[k]ΠΣΠδ), for constants k, δ, have been a source of many great ideas in the last two decades. For eg. depth-3 ideas (Dvir & Shpilka, STOC'05; Kayal & Saxena, CCC'06; Saxena & Seshadhri, FOCS'10, STOC'11); depth-4 ideas (Beecken, Mittmann & Saxena, ICALP'11; Saha, Saxena & Saptharishi, Comput.Compl.'13; Forbes, FOCS'15; Kumar & Saraf, CCC'16); geometric Sylvester-Gallai ideas (Kayal & Saraf, FOCS'09; Shpilka, STOC'19; Peleg & Shpilka, CCC'20, STOC'21). We solve two of the basic underlying open problems in this work. We give the first polynomial-time PIT for (Σ[k]ΠΣ∧). Further, we give the first quasipolynomial time blackbox PIT for both (Σ[k]ΠΣ∧) and (Σ[k]ΠΣΠδ). No subexponential time algorithm was known prior to this work (even if k = δ = 3). A key technical ingredient in all the three algorithms is how the logarithmic derivative, and its power-series, modify the top Π-gate to ∧.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"13 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":"126554981","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":"Toward better depth lower bounds: the XOR-KRW conjecture","authors":"Ivan Mihajlin, A. Smal","doi":"10.4230/LIPIcs.CCC.2021.38","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.38","url":null,"abstract":"In this paper, we propose a new conjecture, the XOR-KRW conjecture, which is a relaxation of the Karchmer-Raz-Wigderson conjecture [10]. This relaxation is still strong enough to imply P ⊈ NC1 if proven. We also present a weaker version of this conjecture that might be used for breaking n3 lower bound for De Morgan formulas. Our study of this conjecture allows us to partially answer an open question stated in [5] regarding the composition of the universal relation with a function. To be more precise, we prove that there exists a function g such that the composition of the universal relation with g is significantly harder than just a universal relation. The fact that we can only prove the existence of g is an inherent feature of our approach. The paper's main technical contribution is a new approach to lower bounds for multiplexer-type relations based on the non-deterministic hardness of non-equality and a new method of converting lower bounds for multiplexer-type relations into lower bounds against some function. In order to do this, we develop techniques to lower bound communication complexity in half-duplex and partially half-duplex communication models.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"4 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133077706","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":"SOS lower bound for exact planted clique","authors":"Shuo Pang","doi":"10.4230/LIPIcs.CCC.2021.26","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.26","url":null,"abstract":"We prove a SOS degree lower bound for the planted clique problem on the Erdös-Rényi random graph G(n, 1/2). The bound we get is degree d = Ω(ϵ2 log n/ log log n) for clique size ω = n1/2−ϵ, which is almost tight. This improves the result of [5] for the \"soft\" version of the problem, where the family of the equality-axioms generated by x1 + ... + xn = ω is relaxed to one inequality x1 + ... + xn ≥ ω. As a technical by-product, we also \"naturalize\" certain techniques that were developed and used for the relaxed problem. This includes a new way to define the pseudo-expectation, and a more robust method to solve out the coarse diagonalization of the moment matrix.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"24 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":"123478822","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":"On the pseudo-deterministic query complexity of NP search problems","authors":"S. Goldwasser, R. Impagliazzo, T. Pitassi, R. Santhanam","doi":"10.4230/LIPIcs.CCC.2021.36","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.36","url":null,"abstract":"We study pseudo-deterministic query complexity - randomized query algorithms that are required to output the same answer with high probability on all inputs. We prove [EQUATION] lower bounds on the pseudo-deterministic complexity of a large family of search problems based on unsatisfiable random CNF instances, and also for the promise problem (FIND1) of finding a 1 in a vector populated with at least half one's. This gives an exponential separation between randomized query complexity and pseudo-deterministic complexity, which is tight in the quantum setting. As applications we partially solve a related combinatorial coloring problem, and we separate random tree-like Resolution from its pseudo-deterministic version. In contrast to our lower bound, we show, surprisingly, that in the zero-error, average case setting, the three notions (deterministic, randomized, pseudo-deterministic) collapse.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"45 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":"116037118","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":"Communication complexity with defective randomness","authors":"Marshall Ball, Oded Goldreich, T. Malkin","doi":"10.4230/LIPIcs.CCC.2021.14","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.14","url":null,"abstract":"Starting with the two standard model of randomized communication complexity, we study the communication complexity of functions when the protocol has access to a defective source of randomness. Specifically, we consider both the public-randomness and private-randomness cases, while replacing the commonly postulated perfect randomness with distributions over ℓ bit strings that have min-entropy at least k ≤ ℓ. We present general upper and lower bounds on the communication complexity in these cases, where the bounds are typically linear in ℓ − k and also depend on the size of the fooling set for the function being computed and on its standard randomized complexity.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"16 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":"114000687","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}