Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing最新文献

{"title":"Indistinguishability obfuscation from circular security","authors":"Romain Gay, R. Pass","doi":"10.1145/3406325.3451070","DOIUrl":"https://doi.org/10.1145/3406325.3451070","url":null,"abstract":"We show the existence of indistinguishability obfuscators (iO) for general circuits assuming subexponential security of: (a) the Learning with Errors (LWE) assumption (with subexponential modulus-to-noise ratio); (b) a circular security conjecture regarding the Gentry-Sahai-Waters' (GSW) encryption scheme and a Packed version of Regev's encryption scheme. The circular security conjecture states that a notion of leakage-resilient security, that we prove is satisfied by GSW assuming LWE, is retained in the presence of an encrypted key-cycle involving GSW and Packed Regev.","PeriodicalId":132752,"journal":{"name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","volume":"49 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121482887","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":"Inverse-exponential correlation bounds and extremely rigid matrices from a new derandomized XOR lemma","authors":"Lijie Chen, Xin Lyu","doi":"10.1145/3406325.3451132","DOIUrl":"https://doi.org/10.1145/3406325.3451132","url":null,"abstract":"In this work we prove that there is a function f ∈ E NP such that, for every sufficiently large n and d = √n/logn, fn (f restricted to n-bit inputs) cannot be (1/2 + 2−d)-approximated by F2-polynomials of degree d. We also observe that a minor improvement (e.g., improving d to n1/2+ε for any ε > 0) over our result would imply E NP cannot be computed by depth-3 AC0-circuits of 2n1/2 + ε size, which is a notoriously hard open question in complexity theory. Using the same proof techniques, we are also able to construct extremely rigid matrices over F2 in P NP. More specifically, we show that for every constant ε ∈ (0,1), there is a P NP algorithm which on input 1n outputs an n× n F2-matrix Hn satisfying RHn(2log1 − ε n) ≥ (1/2 − exp(−log2/3 · ε n) ) · n2, for every sufficiently large n. This improves the recent P NP constructions of rigid matrices in [Alman and Chen, FOCS 2019] and [Bhangale et al., FOCS 2020], which only give Ω(n2) rigidity. The key ingredient in the proof of our new results is a new derandomized XOR lemma based on approximate linear sums, which roughly says that given an n-input function f which cannot be 0.99-approximated by certain linear sum of s many functions in F within ℓ1-distance, one can construct a new function Ampf with O(n) input bits, which cannot be (1/2+sΩ(1))-approximated by F-functions. Taking F to be a function collection containing low-degree F2-polynomials or low-rank F2-matrices, our results are then obtained by first using the algorithmic method to construct a function which is weakly hard against linear sums of F in the above sense, and then applying the derandomized XOR lemma to f. We obtain our new derandomized XOR lemma by giving a generalization of the famous hardcore lemma by Impagliazzo. Our generalization in some sense constructs a non-Boolean hardcore of a weakly hard function f with respect to F-functions, from the weak inapproximability of f by any linear sum of F with bounded ℓp-norm. This generalization recovers the original hardcore lemma by considering the ℓ∞-norm. Surprisingly, when we switch to the ℓ1-norm, we immediately rediscover Levin’s proof of Yao’s XOR Lemma. That is, these first two proofs of Yao’s XOR Lemma can be unified with our new perspective. For proving the correlation bounds, our new derandomized XOR lemma indeed works with the ℓ4/3-norm.","PeriodicalId":132752,"journal":{"name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","volume":"305 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134447014","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":"Explicit uniquely decodable codes for space bounded channels that achieve list-decoding capacity","authors":"Ronen Shaltiel, Jad Silbak","doi":"10.1145/3406325.3451048","DOIUrl":"https://doi.org/10.1145/3406325.3451048","url":null,"abstract":"We consider codes for space bounded channels. This is a model for communication under noise that was introduced by Guruswami and Smith (J. ACM 2016) and lies between the Shannon (random) and Hamming (adversarial) models. In this model, a channel is a space bounded procedure that reads the codeword in one pass, and modifies at most a p fraction of the bits of the codeword. (1) Explicit uniquely decodable codes for space bounded channels: Our main result is that for every 0 ≤ p < 1/4, there exists a constant δ>0 and a uniquely decodable code with rate 1−H(p) for channels with space nδ. This code is explicit (meaning that encoding and decoding are in poly-time). This improves upon previous explicit codes by Guruswami and Smith, and Kopparty, Shaltiel and Silbak (FOCS 2019). Specifically, we obtain the same space and rate as earlier works, even though prior work gave only list-decodable codes (rather than uniquely decodable codes). (2) Complete characterization of the capacity of space bounded channels: Together with a result by Guruswami and Smith showing the impossibility of unique decoding for p ≥ 1/4, our techniques also give a complete characterization of the capacity R(p) of space n1−o(1) channels, specifically: For 0≤p<1/4 R(p)=1-H(p) and for p ≥1/4 R(p)=0. This capacity is strictly larger than the capacity of Hamming channels for every 0 < p < 1/4, and matches the capacity of list decoding, and binary symmetric channels in this range. Curiously, this shows that R(·) is not continuous at p=1/4. Our results are incomparable to recent work on casual channels (these are stronger channels that read the codeword in one pass, but there is no space restriction). The best known codes for casual channels, due to Chen, Jaggi and Langberg (STOC 2015), are shown to exist by the probabilistic method, and no explicit codes are known. A key new ingredient in our construction is a new notion of “evasiveness” of codes, which is concerned with whether a decoding algorithm rejects a word that is obtained when a channel induces few errors to a uniformly chosen (or pseudorandom) string. We use evasiveness (as well as several additional new ideas related to coding theory and pseudorandomness) to identify the “correct” message in the list obtained by previous list-decoding algorithms.","PeriodicalId":132752,"journal":{"name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131064343","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":"Sampling matrices from Harish-Chandra–Itzykson–Zuber densities with applications to Quantum inference and differential privacy","authors":"Jonathan Leake, Colin S. McSwiggen, Nisheeth K. Vishnoi","doi":"10.1145/3406325.3451094","DOIUrl":"https://doi.org/10.1145/3406325.3451094","url":null,"abstract":"Given two Hermitian matrices Y and Λ, the Harish-Chandra–Itzykson–Zuber (HCIZ) distribution is given by the density eTr(U Λ U*Y) with respect to the Haar measure on the unitary group. Random unitary matrices distributed according to the HCIZ distribution are important in various settings in physics and random matrix theory, but the problem of sampling efficiently from this distribution has remained open. We present two algorithms to sample matrices from distributions that are close to the HCIZ distribution. The first produces samples that are ξ-close in total variation distance, and the number of arithmetic operations required depends on poly(log1/ξ). The second produces samples that are ξ-close in infinity divergence, but with a poly(1/ξ) dependence. Our results have the following applications: 1) an efficient algorithm to sample from complex versions of matrix Langevin distributions studied in statistics, 2) an efficient algorithm to sample from continuous maximum entropy distributions over unitary orbits, which in turn implies an efficient algorithm to sample a pure quantum state from the entropy-maximizing ensemble representing a given density matrix, and 3) an efficient algorithm for differentially private rank-k approximation that comes with improved utility bounds for k>1.","PeriodicalId":132752,"journal":{"name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114368717","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":"Decremental all-pairs shortest paths in deterministic near-linear time","authors":"Julia Chuzhoy","doi":"10.1145/3406325.3451025","DOIUrl":"https://doi.org/10.1145/3406325.3451025","url":null,"abstract":"We study the decremental All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. The input to the problem is an undirected n-vertex m-edge graph G with non-negative lengths on edges, that undergoes an online sequence of edge deletions. The goal is to support approximate shortest-paths queries: given a pair x,y of vertices of G, return a path P connecting x to y, whose length is within factor α of the length of the shortest x-y path, in time Õ(|E(P)|), where α is the approximation factor of the algorithm. APSP is one of the most basic and extensively studied dynamic graph problems. A long line of work culminated in the algorithm of [Chechik, FOCS 2018] with near optimal guarantees: for any constant 0<є≤ 1 and parameter k≥ 1, the algorithm achieves approximation factor (2+є)k−1, and total update time O(mn1/k+o(1)log(nL)), where L is the ratio of longest to shortest edge lengths. Unfortunately, as much of prior work, the algorithm is randomized and needs to assume an oblivious adversary; that is, the input edge-deletion sequence is fixed in advance and may not depend on the algorithm’s behavior. In many real-world scenarios, and in applications of APSP to static graph problems, it is crucial that the algorithm works against an adaptive adversary, where the edge deletion sequence may depend on the algorithm’s past behavior arbitrarily; ideally, such an algorithm should be deterministic. Unfortunately, unlike the oblivious-adversary setting, its adaptive-adversary counterpart is still poorly understood. For unweighted graphs, the algorithm of [Henzinger, Krinninger and Nanongkai, FOCS ’13, SICOMP ’16] achieves a (1+є)-approximation with total update time Õ(mn/є); the best current total update time guarantee of n2.5+O(є) is achieved by the recent deterministic algorithm of [Chuzhoy, Saranurak, SODA’21], with 2O(1/є)-multiplicative and 2O(log3/4n/є)-additive approximation. To the best of our knowledge, for arbitrary non-negative edge weights, the fastest current adaptive-update algorithm has total update time O(n3logL/є), achieving a (1+є)-approximation. Even if we are willing to settle for any o(n)-approximation factor, no currently known algorithm has a better than Θ(n3) total update time in weighted graphs and better than Θ(n2.5) total update time in unweighted graphs. Several conditional lower bounds suggest that no algorithm with a sufficiently small approximation factor can achieve an o(n3) total update time. Our main result is a deterministic algorithm for decremental APSP in undirected edge-weighted graphs, that, for any Ω(1/loglogm)≤ є< 1, achieves approximation factor (logm)2O(1/є), with total update time O(m1+O(є)· (logm)O(1/є2)· logL). In particular, we obtain a (polylogm)-approximation in time Õ(m1+є) for any constant є, and, for any slowly growing function f(m), we obtain (logm)f(m)-approximation in time m1+o(1). We also provide an algorithm with similar guarantees for decremental Sparse Neighborhood Covers.","PeriodicalId":132752,"journal":{"name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115013362","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 error resilience of adaptive message exchange","authors":"K. Efremenko, Gillat Kol, Raghuvansh R. Saxena","doi":"10.1145/3406325.3451077","DOIUrl":"https://doi.org/10.1145/3406325.3451077","url":null,"abstract":"We study the error resilience of the message exchange task: Two parties, each holding a private input, want to exchange their inputs. However, the channel connecting them is governed by an adversary that may corrupt a constant fraction of the transmissions. What is the maximum fraction of corruptions that still allows the parties to exchange their inputs? For the non-adaptive channel, where the parties must agree in advance on the order in which they communicate, the maximum error resilience was shown to be 1/4 (see Braverman and Rao, STOC 2011). The problem was also studied over the adaptive channel, where the order in which the parties communicate may not be predetermined (Ghaffari, Haeupler, and Sudan, STOC 2014; Efremenko, Kol, and Saxena, STOC 2020). These works show that the adaptive channel admits much richer set of protocols but leave open the question of finding its maximum error resilience. In this work, we show that the maximum error resilience of a protocol for message exchange over the adaptive channel is 5/16, thereby settling the above question. Our result requires improving both the known upper bounds and the known lower bounds for the problem.","PeriodicalId":132752,"journal":{"name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129910752","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":"Almost optimal super-constant-pass streaming lower bounds for reachability","authors":"Lijie Chen, Gillat Kol, Dmitry Paramonov, Raghuvansh R. Saxena, Zhao Song, Huacheng Yu","doi":"10.1145/3406325.3451038","DOIUrl":"https://doi.org/10.1145/3406325.3451038","url":null,"abstract":"We give an almost quadratic n2−o(1) lower bound on the space consumption of any o(√logn)-pass streaming algorithm solving the (directed) s-t reachability problem. This means that any such algorithm must essentially store the entire graph. As corollaries, we obtain almost quadratic space lower bounds for additional fundamental problems, including maximum matching, shortest path, matrix rank, and linear programming. Our main technical contribution is the definition and construction of set hiding graphs, that may be of independent interest: we give a general way of encoding a set S ⊆ [k] as a directed graph with n = k 1 + o( 1 ) vertices, such that deciding whether i ∈ S boils down to deciding if ti is reachable from si, for a specific pair of vertices (si,ti) in the graph. Furthermore, we prove that our graph “hides” S, in the sense that no low-space streaming algorithm with a small number of passes can learn (almost) anything about S.","PeriodicalId":132752,"journal":{"name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128221141","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":"Chasing convex bodies with linear competitive ratio (invited paper)","authors":"C. Argue, Anupam Gupta, Guru Guruganesh, Ziye Tang","doi":"10.1145/3406325.3465354","DOIUrl":"https://doi.org/10.1145/3406325.3465354","url":null,"abstract":"The problem of chasing convex functions is easy to state: faced with a sequence of convex functions f t over d-dimensional Euclidean spaces, the goal of the algorithm is to output a point x t at each time, so that the sum of the function costs f t (x t ), plus the movement costs ||x t − x t − 1 || is minimized. This problem generalizes questions in online algorithms such as caching and the k-server problem. In 1994, Friedman and Linial posed the question of getting an algorithm with a competitive ratio that depends only on the dimension d. In this talk we give an O (d)-competitive algorithm, based on the notion of the Steiner point of a convex body.","PeriodicalId":132752,"journal":{"name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116625678","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":"Lower bounds for monotone arithmetic circuits via communication complexity","authors":"A. Chattopadhyay, Rajit Datta, P. Mukhopadhyay","doi":"10.1145/3406325.3451069","DOIUrl":"https://doi.org/10.1145/3406325.3451069","url":null,"abstract":"Valiant (1980) showed that general arithmetic circuits with negation can be exponentially more powerful than monotone ones. We give the first improvement to this classical result: we construct a family of polynomials Pn in n variables, each of its monomials has non-negative coefficient, such that Pn can be computed by a polynomial-size depth-three formula but every monotone circuit computing it has size 2Ω(n1/4/log(n)). The polynomial Pn embeds the SINK∘ XOR function devised recently by Chattopadhyay, Mande and Sherif (2020) to refute the Log-Approximate-Rank Conjecture in communication complexity. To prove our lower bound for Pn, we develop a general connection between corruption of combinatorial rectangles by any function f ∘ XOR and corruption of product polynomials by a certain polynomial Pf that is an arithmetic embedding of f. This connection should be of independent interest. Using further ideas from communication complexity, we construct another family of set-multilinear polynomials fn,m such that both Fn,m − є· fn,m and Fn,m + є· fn,m have monotone circuit complexity 2Ω(n/log(n)) if є ≥ 2− Ω( m ) and Fn,m ∏i=1n (xi,1 +⋯+xi,m), with m = O( n/logn ). The polynomials fn,m have 0/1 coefficients and are in VNP. Proving such lower bounds for monotone circuits has been advocated recently by Hrubeš (2020) as a first step towards proving lower bounds against general circuits via his new approach.","PeriodicalId":132752,"journal":{"name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","volume":"51 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134360960","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":"Entropy decay in the Swendsen–Wang dynamics on ℤd","authors":"Antonio Blanca, P. Caputo, D. Parisi, A. Sinclair, Eric Vigoda","doi":"10.1145/3406325.3451095","DOIUrl":"https://doi.org/10.1145/3406325.3451095","url":null,"abstract":"We study the mixing time of the Swendsen-Wang dynamics for the ferromagnetic Ising and Potts models on the integer lattice ℤd. This dynamics is a widely used Markov chain that has largely resisted sharp analysis because it is non-local, i.e., it changes the entire configuration in one step. We prove that, whenever strong spatial mixing (SSM) holds, the mixing time on any n-vertex cube in ℤd is O(logn), and we prove this is tight by establishing a matching lower bound. The previous best known bound was O(n). SSM is a standard condition corresponding to exponential decay of correlations with distance between spins on the lattice and is known to hold in d=2 dimensions throughout the high-temperature (single phase) region. Our result follows from a modified log-Sobolev inequality, which expresses the fact that the dynamics contracts relative entropy at a constant rate at each step. The proof of this fact utilizes a new factorization of the entropy in the joint probability space over spins and edges that underlies the Swendsen-Wang dynamics, which extends to general bipartite graphs of bounded degree. This factorization leads to several additional results, including mixing time bounds for a number of natural local and non-local Markov chains on the joint space, as well as for the standard random-cluster dynamics.","PeriodicalId":132752,"journal":{"name":"Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing","volume":"155 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127352086","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}