{"title":"Junta distance approximation with sub-exponential queries","authors":"Vishnu Iyer, Avishay Tal, Michael Whitmeyer","doi":"10.4230/LIPIcs.CCC.2021.24","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.24","url":null,"abstract":"Leveraging tools of De, Mossel, and Neeman [FOCS, 2019], we show two different results pertaining to the tolerant testing of juntas. Given black-box access to a Boolean function f : {±1}n → {±1}: 1. We give a [EQUATION] query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ + ε)-far from k'-juntas, where [EQUATION]. 2. In the non-relaxed setting, we extend our ideas to give a [EQUATION] (adaptive) query algorithm that distinguishes between functions that are γ-close to k-juntas and (γ + ε)-far from k-juntas. To the best of our knowledge, this is the first subexponential-in-k query algorithm for approximating the distance of f to being a k-junta (previous results of Blais, Canonne, Eden, Levi, and Ron [SODA, 2018] and De, Mossel, and Neeman [FOCS, 2019] required exponentially many queries in k). Our techniques are Fourier analytical and make use of the notion of \"normalized influences\" that was introduced by Talagrand [32].","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121630407","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 simple proof of a new set disjointness with applications to data streams","authors":"Akshay Kamath, Eric Price, David P. Woodruff","doi":"10.4230/LIPIcs.CCC.2021.37","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.37","url":null,"abstract":"The multiplayer promise set disjointness is one of the most widely used problems from communication complexity in applications. In this problem there are k players with subsets S1, ..., Sk, each drawn from {1, 2,..., n}, and we are promised that either the sets are (1) pairwise disjoint, or (2) there is a unique element j occurring in all the sets, which are otherwise pairwise disjoint. The total communication of solving this problem with constant probability in the blackboard model is Ω(n/k). We observe for most applications, it instead suffices to look at what we call the \"mostly\" set disjointness problem, which changes case (2) to say there is a unique element j occurring in at least half of the sets, and the sets are otherwise disjoint. This change gives us a much simpler proof of an Ω(n/k) randomized total communication lower bound, avoiding Hellinger distance and Poincare inequalities. Our proof also gives strong lower bounds for high probability protocols, which are much larger than what is possible for the set disjointness problem. Using this we show several new results for data streams: 1. for ℓ2-Heavy Hitters, any O(1)-pass streaming algorithm in the insertion-only model for detecting if an ε-ℓ2-heavy hitter exists requires [EQUATION] bits of memory, which is optimal up to a log n factor. For deterministic algorithms and constant ε, this gives an Ω(n1/2) lower bound, improving the prior Ω(log n) lower bound. We also obtain lower bounds for Zipfian distributions. 2. for ℓp-Estimation, p > 2, we show an O(1)-pass Ω(n1−2/p log(1/δ)) bit lower bound for outputting an O(1)- approximation with probability 1 − δ, in the insertion-only model. This is optimal, and the best previous lower bound was Ω(n1−2/p + log(1/δ)). 3. for low rank approximation of a sparse matrix in RdXn, if we see the rows of a matrix one at a time in the row-order model, each row having O(1) non-zero entries, any deterministic algorithm requires [EQUATION] memory to output an O(1)-approximate rank-1 approximation. Finally, we consider strict and general turnstile streaming models, and show separations between sketching lower bounds and non-sketching upper bounds for the heavy hitters problem.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115335445","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":"GSF-locality is not sufficient for proximity-oblivious testing","authors":"Isolde Adler, N. Köhler, Pan Peng","doi":"10.4230/LIPIcs.CCC.2021.34","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.34","url":null,"abstract":"In Property Testing, proximity-oblivious testers (POTs) form a class of particularly simple testing algorithms, where a basic test is performed a number of times that may depend on the proximity parameter, but the basic test itself is independent of the proximity parameter. In their seminal work, Goldreich and Ron [STOC 2009; SICOMP 2011] show that the graph properties that allow constant-query proximity-oblivious testing in the bounded-degree model are precisely the properties that can be expressed as a generalised subgraph freeness (GSF) property that satisfies the non-propagation condition. It is left open whether the non-propagation condition is necessary. Indeed, calling properties expressible as a generalised subgraph freeness property GSF-local properties, they ask whether all GSF-local properties are non-propagating. We give a negative answer by exhibiting a property of graphs that is GSF-local and propagating. Hence in particular, our property does not admit a POT, despite being GSF-local. We prove our result by exploiting a recent work of the authors which constructed a first-order (FO) property that is not testable [SODA 2021], and a new connection between FO properties and GSF-local properties via neighbourhood profiles.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129846382","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 stress-free sum-of-squares lower bound for coloring","authors":"Pravesh Kothari, Peter Manohar","doi":"10.4230/LIPIcs.CCC.2021.23","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.23","url":null,"abstract":"We prove that with high probability over the choice of a random graph G from the Erdős-Rényi distribution G(n, 1/2), a natural nO(ε2 log n)-time, degree O(ε2 log n) sum-of-squares semidefinite program cannot refute the existence of a valid k-coloring of G for k = n1/2+ε. Our result implies that the refutation guarantee of the basic semidefinite program (a close variant of the Lovász theta function) cannot be appreciably improved by a natural o(log n)-degree sum-of-squares strengthening, and this is tight up to a no(1) slack in k. To the best of our knowledge, this is the first lower bound for coloring G(n, 1/2) for even a single round strengthening of the basic SDP in any SDP hierarchy. Our proof relies on a new variant of instance-preserving non-pointwise complete reduction within SoS from coloring a graph to finding large independent sets in it. Our proof is (perhaps surprisingly) short, simple and does not require complicated spectral norm bounds on random matrices with dependent entries that have been otherwise necessary in the proofs of many similar results [12, 33, 45, 28, 51]. Our result formally holds for a constraint system where vertices are allowed to belong to multiple color classes; we leave the extension to the formally stronger formulation of coloring, where vertices must belong to unique colors classes, as an outstanding open problem.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"229 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123037330","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":"Variety evasive subspace families","authors":"Zeyu Guo","doi":"10.4230/LIPIcs.CCC.2021.20","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.20","url":null,"abstract":"We introduce the problem of constructing explicit variety evasive subspace families. Given a family F of sub varieties of a projective or affine space, a collection H of projective or affine k-subspaces is (F, ϵ)-evasive if for every V ∈ F, all but at most ϵ-fraction of W ∈ H intersect every irreducible component of V with (at most) the expected dimension. The problem of constructing such an explicit subspace family generalizes both deterministic black-box polynomial identity testing (PIT) and the problem of constructing explicit (weak) lossless rank condensers. Using Chow forms, we construct explicit k-subspace families of polynomial size that are evasive for all varieties of bounded degree in a projective or affine n-space. As one application, we obtain a complete derandomization of Noether's normalization lemma for varieties of bounded degree in a projective or affine n-space. In another application, we obtain a simple polynomial-time black-box PIT algorithm for depth-4 arithmetic circuits with bounded top fan-in and bottom fan-in that are not in the Sylvester-Gallai configuration, improving and simplifying a result of Gupta (ECCC TR 14-130). As a complement of our explicit construction, we prove a lower bound for the size of k-subspace families that are evasive for degree-d varieties in a projective n-space. When n − k = nΩ(1), the lower bound is superpolynomial unless d is bounded. The proof uses a dimension-counting argument on Chow varieties that parametrize projective subvarieties.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123230682","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":"Fourier growth of parity decision trees","authors":"Uma Girish, Avishay Tal, Kewen Wu","doi":"10.4230/LIPIcs.CCC.2021.39","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.39","url":null,"abstract":"We prove that for every parity decision tree of depth d on n variables, the sum of absolute values of Fourier coefficients at level ℓ is at most dℓ/2 · O(ℓ · log(n))ℓ. Our result is nearly tight for small values of ℓ and extends a previous Fourier bound for standard decision trees by Sherstov, Storozhenko, and Wu (STOC, 2021). As an application of our Fourier bounds, using the results of Bansal and Sinha (STOC, 2021), we show that the k-fold Forrelation problem has (randomized) parity decision tree complexity ΩT(n1−1/k), while having quantum query complexity ⌈k/2⌉. Our proof follows a random-walk approach, analyzing the contribution of a random path in the decision tree to the level-ℓ Fourier expression. To carry the argument, we apply a careful cleanup procedure to the parity decision tree, ensuring that the value of the random walk is bounded with high probability. We observe that step sizes for the level-ℓ walks can be computed by the intermediate values of level ≤ ℓ - 1 walks, which calls for an inductive argument. Our approach differs from previous proofs of Tal (FOCS, 2020) and Sherstov, Storozhenko, and Wu (STOC, 2021) that relied on decompositions of the tree. In particular, for the special case of standard decision trees we view our proof as slightly simpler and more intuitive. In addition, we prove a similar bound for noisy decision trees of cost at most d - a model that was recently introduced by Ben-David and Blais (FOCS, 2020).","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"189 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121715720","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":"Separating ABPs and some structured formulas in the non-commutative setting","authors":"Prerona Chatterjee","doi":"10.4230/LIPIcs.CCC.2021.7","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.7","url":null,"abstract":"The motivating question for this work is a long standing open problem, posed by Nisan [20], regarding the relative powers of algebraic branching programs (ABPs) and formulas in the non-commutative setting. Even though the general question remains open, we make some progress towards its resolution. To that effect, we generalise the notion of ordered polynomials in the non-commutative setting (defined by Hrubeš, Wigderson and Yehudayoff [11]) to define abecedarian polynomials and models that naturally compute them. Our main contribution is a possible new approach towards resolving the VFnc vs VBPnc question, via lower bounds against abecedarian formulas. In particular, we show the following. There is an explicit n2-variate degree d abecedarian polynomial fn,d(x) such that • fn,d(X) can be computed by an abecedarian ABP of size O(nd); • any abecedarian formula computing fn,log n(X) must have size at least nΩ(log log n). We also show that a super-polynomial lower bound against abecedarian formulas for flog n, n(X) would separate the powers of formulas and ABPs in the non-commutative setting.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"19 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121968611","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}
Peter Bürgisser, M. Dogan, V. Makam, M. Walter, A. Wigderson
{"title":"Polynomial time algorithms in invariant theory for torus actions","authors":"Peter Bürgisser, M. Dogan, V. Makam, M. Walter, A. Wigderson","doi":"10.4230/LIPIcs.CCC.2021.32","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.32","url":null,"abstract":"An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic \"isomorphism\" or \"classification\" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. These orbit problems extend the more basic null cone problem, whose algorithmic complexity has seen significant progress in recent years. In this paper, we initiate a study of these problems by focusing on the actions of commutative groups (namely, tori). We explain how this setting is motivated from questions in algebraic complexity, and is still rich enough to capture interesting combinatorial algorithmic problems. While the structural theory of commutative actions is well understood, no general efficient algorithms were known for the aforementioned problems. Our main results are polynomial time algorithms for all three problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). Our techniques are based on a combination of fundamental results in invariant theory, linear programming, and algorithmic lattice theory.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114783029","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":"Barriers for recent methods in geodesic optimization","authors":"Cole Franks, Philipp Reichenbach","doi":"10.4230/LIPIcs.CCC.2021.13","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.13","url":null,"abstract":"We study a class of optimization problems including matrix scaling, matrix balancing, multidimensional array scaling, operator scaling, and tensor scaling that arise frequently in theory and in practice. Some of these problems, such as matrix and array scaling, are convex in the Euclidean sense, but others such as operator scaling and tensor scaling are geodesically convex on a different Riemannian manifold. Trust region methods, which include box-constrained Newton's method, are known to produce high precision solutions very quickly for matrix scaling and matrix balancing (Cohen et. al., FOCS 2017, Allen-Zhu et. al. FOCS 2017), and result in polynomial time algorithms for some geodesically convex problems like operator scaling (Garg et. al. STOC 2018, Bürgisser et. al. FOCS 2019). One is led to ask whether these guarantees also hold for multidimensional array scaling and tensor scaling. We show that this is not the case by exhibiting instances with exponential diameter bound: we construct polynomial-size instances of 3-dimensional array scaling and 3-tensor scaling whose approximate solutions all have doubly exponential condition number. Moreover, we study convex-geometric notions of complexity known as margin and gap, which are used to bound the running times of all existing optimization algorithms for such problems. We show that margin and gap are exponentially small for several problems including array scaling, tensor scaling and polynomial scaling. Our results suggest that it is impossible to prove polynomial running time bounds for tensor scaling based on diameter bounds alone. Therefore, our work motivates the search for analogues of more sophisticated algorithms, such as interior point methods, for geodesically convex optimization that do not rely on polynomial diameter bounds.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128760167","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}
Noah Fleming, Mika Göös, R. Impagliazzo, T. Pitassi, Robert Robere, Li-Yang Tan, A. Wigderson
{"title":"On the power and limitations of branch and cut","authors":"Noah Fleming, Mika Göös, R. Impagliazzo, T. Pitassi, Robert Robere, Li-Yang Tan, A. Wigderson","doi":"10.4230/LIPIcs.CCC.2021.6","DOIUrl":"https://doi.org/10.4230/LIPIcs.CCC.2021.6","url":null,"abstract":"The Stabbing Planes proof system [8] was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas - certain unsatisfiable systems of linear equations mod2 - which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari [25] showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes. In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients (SP*) can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas.","PeriodicalId":336911,"journal":{"name":"Proceedings of the 36th Computational Complexity Conference","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134436772","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}