Jarosław Błasiok, V. Guruswami, Preetum Nakkiran, A. Rudra, M. Sudan
Arikan’s exciting discovery of polar codes has provided an altogether new way to efficiently achieve Shannon capacity. Given a (constant-sized) invertible matrix M, a family of polar codes can be associated with this matrix and its ability to approach capacity follows from the polarization of an associated [0,1]-bounded martingale, namely its convergence in the limit to either 0 or 1 with probability 1. Arikan showed appropriate polarization of the martingale associated with the matrix G2 = ( [complex formula not displayed] ) to get capacity achieving codes. His analysis was later extended to all matrices M which satisfy an obvious necessary condition for polarization. While Arikan’s theorem does not guarantee that the codes achieve capacity at small blocklengths (specifically in length which is a polynomial in 1/є where є is the difference between the capacity of a channel and the rate of the code), it turns out that a “strong” analysis of the polarization of the underlying martingale would lead to such constructions. Indeed for the martingale associated with G2 such a strong polarization was shown in two independent works ([Guruswami and Xia, IEEE IT ’15] and [Hassani et al., IEEE IT’14]), thereby resolving a major theoretical challenge associated with the efficient attainment of Shannon capacity. In this work we extend the result above to cover martingales associated with all matrices that satisfy the necessary condition for (weak) polarization. In addition to being vastly more general, our proofs of strong polarization are (in our view) also much simpler and modular. Key to our proof is a notion of local polarization that only depends on the evolution of the martingale in a single time step. We show that local polarization always implies strong polarization. We then apply relatively simple reasoning about conditional entropies to prove local polarization in very general settings. Specifically, our result shows strong polarization over all prime fields and leads to efficient capacity-achieving source codes for compressing arbitrary i.i.d. sources, and capacity-achieving channel codes for arbitrary symmetric memoryless channels.
阿里坎令人兴奋的极性码的发现提供了一种全新的方法来有效地实现香农容量。给定一个(常数大小的)可逆矩阵M，可以与该矩阵关联一组极码，其接近容量的能力来自于关联的[0,1]有界鞅的极化，即它在极限收敛于0或1的概率为1。Arikan对矩阵G2 =([复公式未显示])相关的鞅进行适当极化，得到容量实现码。他的分析后来推广到所有满足一个明显的极化必要条件的矩阵M。虽然Arikan定理并不能保证代码在小块长度下实现容量(特别是长度是1/ k的多项式，其中k是信道容量和代码速率之间的差)，但事实证明，对底层鞅的极化的“强”分析将导致这样的结构。事实上，对于与G2相关的鞅，这种强烈的极化在两个独立的著作([Guruswami and Xia, IEEE IT ' 15]和[Hassani et al.， IEEE IT ' 14])中得到了证明，从而解决了与有效实现香农容量相关的主要理论挑战。在这项工作中，我们扩展了上述结果，以涵盖与满足(弱)极化必要条件的所有矩阵相关的鞅。除了更通用之外，我们的强极化证明(在我们看来)也更简单和模块化。我们证明的关键是局部极化的概念，它只依赖于鞅在单个时间步长的演化。我们证明了局部极化总是意味着强极化。然后，我们应用相对简单的关于条件熵的推理来证明在非常一般的情况下的局部极化。具体地说，我们的结果显示了所有素场上的强极化，并导致压缩任意i.i.d源的有效容量实现源代码和压缩任意对称无内存信道的容量实现信道代码。
{"title":"General strong polarization","authors":"Jarosław Błasiok, V. Guruswami, Preetum Nakkiran, A. Rudra, M. Sudan","doi":"10.1145/3188745.3188816","DOIUrl":"https://doi.org/10.1145/3188745.3188816","url":null,"abstract":"Arikan’s exciting discovery of polar codes has provided an altogether new way to efficiently achieve Shannon capacity. Given a (constant-sized) invertible matrix M, a family of polar codes can be associated with this matrix and its ability to approach capacity follows from the polarization of an associated [0,1]-bounded martingale, namely its convergence in the limit to either 0 or 1 with probability 1. Arikan showed appropriate polarization of the martingale associated with the matrix G2 = ( [complex formula not displayed] ) to get capacity achieving codes. His analysis was later extended to all matrices M which satisfy an obvious necessary condition for polarization. While Arikan’s theorem does not guarantee that the codes achieve capacity at small blocklengths (specifically in length which is a polynomial in 1/є where є is the difference between the capacity of a channel and the rate of the code), it turns out that a “strong” analysis of the polarization of the underlying martingale would lead to such constructions. Indeed for the martingale associated with G2 such a strong polarization was shown in two independent works ([Guruswami and Xia, IEEE IT ’15] and [Hassani et al., IEEE IT’14]), thereby resolving a major theoretical challenge associated with the efficient attainment of Shannon capacity. In this work we extend the result above to cover martingales associated with all matrices that satisfy the necessary condition for (weak) polarization. In addition to being vastly more general, our proofs of strong polarization are (in our view) also much simpler and modular. Key to our proof is a notion of local polarization that only depends on the evolution of the martingale in a single time step. We show that local polarization always implies strong polarization. We then apply relatively simple reasoning about conditional entropies to prove local polarization in very general settings. Specifically, our result shows strong polarization over all prime fields and leads to efficient capacity-achieving source codes for compressing arbitrary i.i.d. sources, and capacity-achieving channel codes for arbitrary symmetric memoryless channels.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"61 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84570942","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}
The approximate degree of a Boolean function f(x1,x2,…,xn) is the minimum degree of a real polynomial that approximates f pointwise within 1/3. Upper bounds on approximate degree have a variety of applications in learning theory, differential privacy, and algorithm design in general. Nearly all known upper bounds on approximate degree arise in an existential manner from bounds on quantum query complexity. We develop a first-principles, classical approach to the polynomial approximation of Boolean functions. We use it to give the first constructive upper bounds on the approximate degree of several fundamental problems: (i) O(n3/4−1/(4(2k−1))) for the k-element distinctness problem; (ii) O(n1−1/(k+1)) for the k-subset sum problem; (iii) O(n1−1/(k+1)) for any k-DNF or k-CNF formula; (iv) O(n3/4) for the surjectivity problem. In all cases, we obtain explicit, closed-form approximating polynomials that are unrelated to the quantum arguments from previous work. Our first three results match the bounds from quantum query complexity. Our fourth result improves polynomially on the Θ(n) quantum query complexity of the problem and refutes the conjecture by several experts that surjectivity has approximate degree Ω(n). In particular, we exhibit the first natural problem with a polynomial gap between approximate degree and quantum query complexity.
{"title":"Algorithmic polynomials","authors":"Alexander A. Sherstov","doi":"10.1145/3188745.3188958","DOIUrl":"https://doi.org/10.1145/3188745.3188958","url":null,"abstract":"The approximate degree of a Boolean function f(x1,x2,…,xn) is the minimum degree of a real polynomial that approximates f pointwise within 1/3. Upper bounds on approximate degree have a variety of applications in learning theory, differential privacy, and algorithm design in general. Nearly all known upper bounds on approximate degree arise in an existential manner from bounds on quantum query complexity. We develop a first-principles, classical approach to the polynomial approximation of Boolean functions. We use it to give the first constructive upper bounds on the approximate degree of several fundamental problems: (i) O(n3/4−1/(4(2k−1))) for the k-element distinctness problem; (ii) O(n1−1/(k+1)) for the k-subset sum problem; (iii) O(n1−1/(k+1)) for any k-DNF or k-CNF formula; (iv) O(n3/4) for the surjectivity problem. In all cases, we obtain explicit, closed-form approximating polynomials that are unrelated to the quantum arguments from previous work. Our first three results match the bounds from quantum query complexity. Our fourth result improves polynomially on the Θ(n) quantum query complexity of the problem and refutes the conjecture by several experts that surjectivity has approximate degree Ω(n). In particular, we exhibit the first natural problem with a polynomial gap between approximate degree and quantum query complexity.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"290 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-01-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79457747","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}
The completely positive maps, a generalization of the nonnegative matrices, are a well-studied class of maps from n× n matrices to m× m matrices. The existence of the operator analogues of doubly stochastic scalings of matrices, the study of which is known as operator scaling, is equivalent to a multitude of problems in computer science and mathematics such rational identity testing in non-commuting variables, noncommutative rank of symbolic matrices, and a basic problem in invariant theory (Garg et. al., 2016). We study operator scaling with specified marginals, which is the operator analogue of scaling matrices to specified row and column sums (or marginals). We characterize the operators which can be scaled to given marginals, much in the spirit of the Gurvits’ algorithmic characterization of the operators that can be scaled to doubly stochastic (Gurvits, 2004). Our algorithm, which is a modified version of Gurvits’ algorithm, produces approximate scalings in time poly(n,m) whenever scalings exist. A central ingredient in our analysis is a reduction from operator scaling with specified marginals to operator scaling in the doubly stochastic setting. Instances of operator scaling with specified marginals arise in diverse areas of study such as the Brascamp-Lieb inequalities, communication complexity, eigenvalues of sums of Hermitian matrices, and quantum information theory. Some of the known theorems in these areas, several of which had no algorithmic proof, are straightforward consequences of our characterization theorem. For instance, we obtain a simple algorithm to find, when it exists, a tuple of Hermitian matrices with given spectra whose sum has a given spectrum. We also prove new theorems such as a generalization of Forster’s theorem (Forster, 2002) concerning radial isotropic position.
{"title":"Operator scaling with specified marginals","authors":"Cole Franks","doi":"10.1145/3188745.3188932","DOIUrl":"https://doi.org/10.1145/3188745.3188932","url":null,"abstract":"The completely positive maps, a generalization of the nonnegative matrices, are a well-studied class of maps from n× n matrices to m× m matrices. The existence of the operator analogues of doubly stochastic scalings of matrices, the study of which is known as operator scaling, is equivalent to a multitude of problems in computer science and mathematics such rational identity testing in non-commuting variables, noncommutative rank of symbolic matrices, and a basic problem in invariant theory (Garg et. al., 2016). We study operator scaling with specified marginals, which is the operator analogue of scaling matrices to specified row and column sums (or marginals). We characterize the operators which can be scaled to given marginals, much in the spirit of the Gurvits’ algorithmic characterization of the operators that can be scaled to doubly stochastic (Gurvits, 2004). Our algorithm, which is a modified version of Gurvits’ algorithm, produces approximate scalings in time poly(n,m) whenever scalings exist. A central ingredient in our analysis is a reduction from operator scaling with specified marginals to operator scaling in the doubly stochastic setting. Instances of operator scaling with specified marginals arise in diverse areas of study such as the Brascamp-Lieb inequalities, communication complexity, eigenvalues of sums of Hermitian matrices, and quantum information theory. Some of the known theorems in these areas, several of which had no algorithmic proof, are straightforward consequences of our characterization theorem. For instance, we obtain a simple algorithm to find, when it exists, a tuple of Hermitian matrices with given spectra whose sum has a given spectrum. We also prove new theorems such as a generalization of Forster’s theorem (Forster, 2002) concerning radial isotropic position.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88710893","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}
In this paper we study the complexity of constructing a hitting set for VP, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given n,s,r in unary outputs a set of rational n-tuples of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all n-variate polynomials of degree r that are the limit of size s algebraic circuits. Previously it was known that a random set of this size is a hitting set, but a construction that is certified to work was only known in EXPSPACE (or EXPH assuming the generalized Riemann hypothesis). As a corollary we get that a host of other algebraic problems such as Noether Normalization Lemma, can also be solved in PSPACE deterministically, where earlier only randomized algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann hypothesis) were known. The proof relies on the new notion of a robust hitting set which is a set of inputs such that any nonzero polynomial that can be computed by a polynomial size algebraic circuit, evaluates to a not too small value on at least one element of the set. Proving the existence of such a robust hitting set is the main technical difficulty in the proof. Our proof uses anti-concentration results for polynomials, basic tools from algebraic geometry and the existential theory of the reals.
{"title":"A PSPACE construction of a hitting set for the closure of small algebraic circuits","authors":"Michael A. Forbes, Amir Shpilka","doi":"10.1145/3188745.3188792","DOIUrl":"https://doi.org/10.1145/3188745.3188792","url":null,"abstract":"In this paper we study the complexity of constructing a hitting set for VP, the class of polynomials that can be infinitesimally approximated by polynomials that are computed by polynomial sized algebraic circuits, over the real or complex numbers. Specifically, we show that there is a PSPACE algorithm that given n,s,r in unary outputs a set of rational n-tuples of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all n-variate polynomials of degree r that are the limit of size s algebraic circuits. Previously it was known that a random set of this size is a hitting set, but a construction that is certified to work was only known in EXPSPACE (or EXPH assuming the generalized Riemann hypothesis). As a corollary we get that a host of other algebraic problems such as Noether Normalization Lemma, can also be solved in PSPACE deterministically, where earlier only randomized algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann hypothesis) were known. The proof relies on the new notion of a robust hitting set which is a set of inputs such that any nonzero polynomial that can be computed by a polynomial size algebraic circuit, evaluates to a not too small value on at least one element of the set. Proving the existence of such a robust hitting set is the main technical difficulty in the proof. Our proof uses anti-concentration results for polynomials, basic tools from algebraic geometry and the existential theory of the reals.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"161 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85415254","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}
Computing shortest paths is one of the central problems in the theory of distributed computing. For the last few years, substantial progress has been made on the approximate single source shortest paths problem, culminating in an algorithm of Henzinger, Krinninger, and Nanongkai [STOC’16] which deterministically computes (1+o(1))-approximate shortest paths in Õ(D+√n) time, where D is the hop-diameter of the graph. Up to logarithmic factors, this time complexity is optimal, matching the lower bound of Elkin [STOC’04]. The question of exact shortest paths however saw no algorithmic progress for decades, until the recent breakthrough of Elkin [STOC’17], which established a sublinear-time algorithm for exact single source shortest paths on undirected graphs. Shortly after, Huang et al. [FOCS’17] provided improved algorithms for exact all pairs shortest paths problem on directed graphs. In this paper, we provide an alternative single-source shortest path algorithm with complexity Õ(n3/4D1/4). For polylogarithmic D, this improves on Elkin’s Õ(n5/6) bound and gets closer to the Ω(n1/2) lower bound of Elkin [STOC’04]. For larger values of D, we present an improved variant of our algorithm which achieves complexity Õ(max{ n3/4+o(1) , n3/4D1/6} + D ), and thus compares favorably with Elkin’s bound of Õ(max{ n5/6, n2/3D1/3} + D ) in essentially the entire range of parameters. This algorithm provides also a qualitative improvement, because it works for the more challenging case of directed graph (i.e., graphs where the two directions of an edge can have different weights), constituting the first sublinear-time algorithm for directed graphs. Our algorithm also extends to the case of exact r-source shortest paths, in which we provide the fastest algorithm for moderately small r and D, improving on those of Huang et al.
{"title":"Improved distributed algorithms for exact shortest paths","authors":"M. Ghaffari, Jason Li","doi":"10.1145/3188745.3188948","DOIUrl":"https://doi.org/10.1145/3188745.3188948","url":null,"abstract":"Computing shortest paths is one of the central problems in the theory of distributed computing. For the last few years, substantial progress has been made on the approximate single source shortest paths problem, culminating in an algorithm of Henzinger, Krinninger, and Nanongkai [STOC’16] which deterministically computes (1+o(1))-approximate shortest paths in Õ(D+√n) time, where D is the hop-diameter of the graph. Up to logarithmic factors, this time complexity is optimal, matching the lower bound of Elkin [STOC’04]. The question of exact shortest paths however saw no algorithmic progress for decades, until the recent breakthrough of Elkin [STOC’17], which established a sublinear-time algorithm for exact single source shortest paths on undirected graphs. Shortly after, Huang et al. [FOCS’17] provided improved algorithms for exact all pairs shortest paths problem on directed graphs. In this paper, we provide an alternative single-source shortest path algorithm with complexity Õ(n3/4D1/4). For polylogarithmic D, this improves on Elkin’s Õ(n5/6) bound and gets closer to the Ω(n1/2) lower bound of Elkin [STOC’04]. For larger values of D, we present an improved variant of our algorithm which achieves complexity Õ(max{ n3/4+o(1) , n3/4D1/6} + D ), and thus compares favorably with Elkin’s bound of Õ(max{ n5/6, n2/3D1/3} + D ) in essentially the entire range of parameters. This algorithm provides also a qualitative improvement, because it works for the more challenging case of directed graph (i.e., graphs where the two directions of an edge can have different weights), constituting the first sublinear-time algorithm for directed graphs. Our algorithm also extends to the case of exact r-source shortest paths, in which we provide the fastest algorithm for moderately small r and D, improving on those of Huang et al.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"6 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72734494","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}
Logarithmic Sobolev inequalities are a powerful way to estimate the rate of convergence of Markov chains and to derive concentration inequalities on distributions. We prove that the log-Sobolev constant of any isotropic logconcave density in Rn with support of diameter D is Ω(1/D), resolving a question posed by Frieze and Kannan in 1997. This is asymptotically the best possible estimate and improves on the previous bound of Ω(1/D2) by Kannan-Lovász-Montenegro. It follows that for any isotropic logconcave density, the ball walk with step size δ=Θ(1/√n) mixes in O*(n2D) proper steps from any starting point. This improves on the previous best bound of O*(n2D2) and is also asymptotically tight. The new bound leads to the following refined large deviation inequality for an L-Lipschitz function g over an isotropic logconcave density p: for any t>0, [complex formula not displayed] where ḡ is the median or mean of g for x∼ p; this improves on previous bounds by Paouris and by Guedon-Milman. Our main proof is based on stochastic localization together with a Stieltjes-type barrier function.
{"title":"Stochastic localization + Stieltjes barrier = tight bound for log-Sobolev","authors":"Y. Lee, S. Vempala","doi":"10.1145/3188745.3188866","DOIUrl":"https://doi.org/10.1145/3188745.3188866","url":null,"abstract":"Logarithmic Sobolev inequalities are a powerful way to estimate the rate of convergence of Markov chains and to derive concentration inequalities on distributions. We prove that the log-Sobolev constant of any isotropic logconcave density in Rn with support of diameter D is Ω(1/D), resolving a question posed by Frieze and Kannan in 1997. This is asymptotically the best possible estimate and improves on the previous bound of Ω(1/D2) by Kannan-Lovász-Montenegro. It follows that for any isotropic logconcave density, the ball walk with step size δ=Θ(1/√n) mixes in O*(n2D) proper steps from any starting point. This improves on the previous best bound of O*(n2D2) and is also asymptotically tight. The new bound leads to the following refined large deviation inequality for an L-Lipschitz function g over an isotropic logconcave density p: for any t>0, [complex formula not displayed] where ḡ is the median or mean of g for x∼ p; this improves on previous bounds by Paouris and by Guedon-Milman. Our main proof is based on stochastic localization together with a Stieltjes-type barrier function.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"51 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87846042","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}
We prove the following quantitative hardness results for the Shortest Vector Problem in the ℓp norm (SVP_p), where n is the rank of the input lattice. For “almost all” p > p0 ≈ 2.1397, there is no 2n/Cp-time algorithm for SVP_p for some explicit (easily computable) constant Cp > 0 unless the (randomized) Strong Exponential Time Hypothesis (SETH) is false. (E.g., for p ≥ 3, Cp < 1 + (p+3) 2−p + 10 p2 2−2p.) For any 1 ≤ p ≤ ∞, there is no 2o(n)-time algorithm for SVP_p unless the non-uniform Gap-Exponential Time Hypothesis (Gap-ETH) is false. Furthermore, for each such p, there exists a constant γp > 1 such that the same result holds even for γp-approximate SVP_p. For p > 2, the above statement holds under the weaker assumption of randomized Gap-ETH. I.e., there is no 2o(n)-time algorithm for γp-approximate SVP_p unless randomized Gap-ETH is false. See http://arxiv.org/abs/1712.00942 for a complete exposition.
{"title":"(Gap/S)ETH hardness of SVP","authors":"Divesh Aggarwal, Noah Stephens-Davidowitz","doi":"10.1145/3188745.3188840","DOIUrl":"https://doi.org/10.1145/3188745.3188840","url":null,"abstract":"We prove the following quantitative hardness results for the Shortest Vector Problem in the ℓp norm (SVP_p), where n is the rank of the input lattice. For “almost all” p > p0 ≈ 2.1397, there is no 2n/Cp-time algorithm for SVP_p for some explicit (easily computable) constant Cp > 0 unless the (randomized) Strong Exponential Time Hypothesis (SETH) is false. (E.g., for p ≥ 3, Cp < 1 + (p+3) 2−p + 10 p2 2−2p.) For any 1 ≤ p ≤ ∞, there is no 2o(n)-time algorithm for SVP_p unless the non-uniform Gap-Exponential Time Hypothesis (Gap-ETH) is false. Furthermore, for each such p, there exists a constant γp > 1 such that the same result holds even for γp-approximate SVP_p. For p > 2, the above statement holds under the weaker assumption of randomized Gap-ETH. I.e., there is no 2o(n)-time algorithm for γp-approximate SVP_p unless randomized Gap-ETH is false. See http://arxiv.org/abs/1712.00942 for a complete exposition.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"3 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84109845","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}
Our first theorem in this paper is a hierarchy theorem for the query complexity of testing graph properties with 1-sided error; more precisely, we show that for every sufficiently fast-growing function f, there is a graph property whose 1-sided-error query complexity is precisely f(Θ(1/ε)). No result of this type was previously known for any f which is super-polynomial. Goldreich [ECCC 2005] asked to exhibit a graph property whose query complexity is 2Θ(1/ε). Our hierarchy theorem partially resolves this problem by exhibiting a property whose 1-sided-error query complexity is 2Θ(1/ε). We also use our hierarchy theorem in order to resolve a problem raised by the second author and Alon [STOC 2005] regarding testing relaxed versions of bipartiteness. Our second theorem states that for any function f there is a graph property whose 1-sided-error query complexity is f(Θ(1/ε)) while its 2-sided-error query complexity is only poly(1/ε). This is the first indication of the surprising power that 2-sided-error testing algorithms have over 1-sided-error ones, even when restricted to properties that are testable with 1-sided error. Again, no result of this type was previously known for any f that is super polynomial. The above theorems are derived from a graph theoretic result which we think is of independent interest, and might have further applications. Alon and Shikhelman [JCTB 2016] introduced the following generalized Turán problem: for fixed graphs H and T, and an integer n, what is the maximum number of copies of T, denoted by ex(n,T,H), that can appear in an n-vertex H-free graph? This problem received a lot of attention recently, with an emphasis on ex(n,C3,C2ℓ +1). Our third theorem in this paper gives tight bounds for ex(n,Ck,Cℓ) for all the remaining values of k and ℓ.
本文的第一个定理是具有单侧误差的图属性测试查询复杂度的层次定理;更准确地说，我们证明了对于每一个足够快速的函数f，存在一个图属性，其单边错误查询复杂度恰好是f(Θ(1/ε))。对于任何超多项式f，以前都不知道这种类型的结果。Goldreich [ECCC 2005]要求展示一个查询复杂度为2Θ(1/ε)的图属性。我们的层次定理通过展示一个单侧错误查询复杂度为2Θ(1/ε)的属性，部分地解决了这个问题。我们还使用我们的层次定理来解决由第二作者和Alon [STOC 2005]提出的关于测试放宽版本的双方性的问题。我们的第二个定理表明，对于任何函数f，存在一个图属性，其单边错误查询复杂度为f(Θ(1/ε))，而其双向错误查询复杂度仅为poly(1/ε)。这是双侧错误测试算法比单侧错误测试算法的惊人威力的第一个迹象，即使局限于可以用单侧错误测试的属性。同样，对于任何f是超多项式的情况，以前都不知道这种类型的结果。以上定理是由图论的一个结果推导出来的，我们认为这个结果有独立的意义，并且可能有进一步的应用。Alon和Shikhelman [JCTB 2016]引入了以下广义Turán问题:对于固定图H和T，以及整数n，在一个n顶点的无H图中，T的最大副本数(用ex(n,T,H)表示)是多少?这个问题最近受到了很多关注，重点是ex(n,C3,C2, r +1)。本文的第三个定理对于k和r的所有剩余值给出了ex(n,Ck,C, r)的紧界。
{"title":"A generalized Turán problem and its applications","authors":"Lior Gishboliner, A. Shapira","doi":"10.1145/3188745.3188778","DOIUrl":"https://doi.org/10.1145/3188745.3188778","url":null,"abstract":"Our first theorem in this paper is a hierarchy theorem for the query complexity of testing graph properties with 1-sided error; more precisely, we show that for every sufficiently fast-growing function f, there is a graph property whose 1-sided-error query complexity is precisely f(Θ(1/ε)). No result of this type was previously known for any f which is super-polynomial. Goldreich [ECCC 2005] asked to exhibit a graph property whose query complexity is 2Θ(1/ε). Our hierarchy theorem partially resolves this problem by exhibiting a property whose 1-sided-error query complexity is 2Θ(1/ε). We also use our hierarchy theorem in order to resolve a problem raised by the second author and Alon [STOC 2005] regarding testing relaxed versions of bipartiteness. Our second theorem states that for any function f there is a graph property whose 1-sided-error query complexity is f(Θ(1/ε)) while its 2-sided-error query complexity is only poly(1/ε). This is the first indication of the surprising power that 2-sided-error testing algorithms have over 1-sided-error ones, even when restricted to properties that are testable with 1-sided error. Again, no result of this type was previously known for any f that is super polynomial. The above theorems are derived from a graph theoretic result which we think is of independent interest, and might have further applications. Alon and Shikhelman [JCTB 2016] introduced the following generalized Turán problem: for fixed graphs H and T, and an integer n, what is the maximum number of copies of T, denoted by ex(n,T,H), that can appear in an n-vertex H-free graph? This problem received a lot of attention recently, with an emphasis on ex(n,C3,C2ℓ +1). Our third theorem in this paper gives tight bounds for ex(n,Ck,Cℓ) for all the remaining values of k and ℓ.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-12-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74692540","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}
S. KarthikC., Bundit Laekhanukit, Pasin Manurangsi
We study the parameterized complexity of approximating the k-Dominating Set (domset) problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a dominating set of size at most F(k) · k whenever the graph G has a dominating set of size k. When such an algorithm runs in time T(k)poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for k-domset. Whether such an algorithm exists is listed in the seminal book of Downey and Fellows (2013) as one of the ”most infamous” open problems in Parameterized Complexity. This work gives an almost complete answer to this question by showing the non-existence of such an algorithm under W[1]≠FPT and further providing tighter running time lower bounds under stronger hypotheses. Specifically, we prove the following for every computable functions T, F and every constant ε > 0: (i) Assuming W[1]≠FPT, there is no F(k)-FPT-approximation algorithm for k-domset, (ii) Assuming the Exponential Time Hypothesis (ETH), there is no F(k)-approximation algorithm for k-domset that runs in T(k)no(k) time, (iii) Assuming the Strong Exponential Time Hypothesis (SETH), for every integer k ≥ 2, there is no F(k)-approximation algorithm for k-domset that runs in T(k)nk − ε time, (iv) Assuming the k-sum Hypothesis, for every integer k ≥ 3, there is no F(k)-approximation algorithm for k-domset that runs in T(k) n⌈ k/2 ⌉ − ε time. Previously, only constant ratio FPT-approximation algorithms were ruled out under W[1]≠FPT and (log1/4 − ε k)-FPT-approximation algorithms were ruled out under ETH [Chen and Lin, FOCS 2016]. Recently, the non-existence of an F(k)-FPT-approximation algorithm for any function F was shown under gapETH [Chalermsook et al., FOCS 2017]. Note that, to the best of our knowledge, no running time lower bound of the form nδ k for any absolute constant δ > 0 was known before even for any constant factor inapproximation ratio. Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well studied problem or a variant of one; this allows us to easily apply known techniques to solve them.
{"title":"On the parameterized complexity of approximating dominating set","authors":"S. KarthikC., Bundit Laekhanukit, Pasin Manurangsi","doi":"10.1145/3188745.3188896","DOIUrl":"https://doi.org/10.1145/3188745.3188896","url":null,"abstract":"We study the parameterized complexity of approximating the k-Dominating Set (domset) problem where an integer k and a graph G on n vertices are given as input, and the goal is to find a dominating set of size at most F(k) · k whenever the graph G has a dominating set of size k. When such an algorithm runs in time T(k)poly(n) (i.e., FPT-time) for some computable function T, it is said to be an F(k)-FPT-approximation algorithm for k-domset. Whether such an algorithm exists is listed in the seminal book of Downey and Fellows (2013) as one of the ”most infamous” open problems in Parameterized Complexity. This work gives an almost complete answer to this question by showing the non-existence of such an algorithm under W[1]≠FPT and further providing tighter running time lower bounds under stronger hypotheses. Specifically, we prove the following for every computable functions T, F and every constant ε > 0: (i) Assuming W[1]≠FPT, there is no F(k)-FPT-approximation algorithm for k-domset, (ii) Assuming the Exponential Time Hypothesis (ETH), there is no F(k)-approximation algorithm for k-domset that runs in T(k)no(k) time, (iii) Assuming the Strong Exponential Time Hypothesis (SETH), for every integer k ≥ 2, there is no F(k)-approximation algorithm for k-domset that runs in T(k)nk − ε time, (iv) Assuming the k-sum Hypothesis, for every integer k ≥ 3, there is no F(k)-approximation algorithm for k-domset that runs in T(k) n⌈ k/2 ⌉ − ε time. Previously, only constant ratio FPT-approximation algorithms were ruled out under W[1]≠FPT and (log1/4 − ε k)-FPT-approximation algorithms were ruled out under ETH [Chen and Lin, FOCS 2016]. Recently, the non-existence of an F(k)-FPT-approximation algorithm for any function F was shown under gapETH [Chalermsook et al., FOCS 2017]. Note that, to the best of our knowledge, no running time lower bound of the form nδ k for any absolute constant δ > 0 was known before even for any constant factor inapproximation ratio. Our results are obtained by establishing a connection between communication complexity and hardness of approximation, generalizing the ideas from a recent breakthrough work of Abboud et al. [FOCS 2017]. Specifically, we show that to prove hardness of approximation of a certain parameterized variant of the label cover problem, it suffices to devise a specific protocol for a communication problem that depends on which hypothesis we rely on. Each of these communication problems turns out to be either a well studied problem or a variant of one; this allows us to easily apply known techniques to solve them.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"34 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91022715","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}
We use the Sum of Squares method to develop new efficient algorithms for learning well-separated mixtures of Gaussians and robust mean estimation, both in high dimensions, that substantially improve upon the statistical guarantees achieved by previous efficient algorithms. Our contributions are: Mixture models with separated means: We study mixtures of poly(k)-many k-dimensional distributions where the means of every pair of distributions are separated by at least kε. In the special case of spherical Gaussian mixtures, we give a kO(1/ε)-time algorithm that learns the means assuming separation at least kε, for any ε> 0. This is the first algorithm to improve on greedy (“single-linkage”) and spectral clustering, breaking a long-standing barrier for efficient algorithms at separation k1/4. Robust estimation: When an unknown (1−ε)-fraction of X1,…,Xn are chosen from a sub-Gaussian distribution with mean µ but the remaining points are chosen adversarially, we give an algorithm recovering µ to error ε1−1/t in time kO(t), so long as sub-Gaussian-ness up to O(t) moments can be certified by a Sum of Squares proof. This is the first polynomial-time algorithm with guarantees approaching the information-theoretic limit for non-Gaussian distributions. Previous algorithms could not achieve error better than ε1/2. As a corollary, we achieve similar results for robust covariance estimation. Both of these results are based on a unified technique. Inspired by recent algorithms of Diakonikolas et al. in robust statistics, we devise an SDP based on the Sum of Squares method for the following setting: given X1,…,Xn ∈ ℝk for large k and n = poly(k) with the promise that a subset of X1,…,Xn were sampled from a probability distribution with bounded moments, recover some information about that distribution.
我们使用平方和方法来开发新的高效算法，用于学习良好分离的高斯混合和鲁棒平均估计，两者都是在高维上，大大提高了以前高效算法所实现的统计保证。我们的贡献是:具有分离均值的混合模型:我们研究多(k)-许多k维分布的混合，其中每对分布的均值至少相隔kε。在球形高斯混合的特殊情况下，我们给出了一个kO(1/ε)时间算法，该算法学习了假设分离至少为kε的均值，对于任何ε> 0。这是第一个改进贪婪(“单链接”)和谱聚类的算法，打破了在分离k1/4时高效算法的长期障碍。鲁棒性估计:当从均值为μ的亚高斯分布中选取未知的(1−ε)分数X1，…，Xn，而其余的点都是逆向选取时，我们给出了一种在kO(t)时间内恢复μ to误差ε1−1/t的算法，只要在O(t)阶矩以内的亚高斯性可以通过平方和证明得到证明。这是第一个多项式时间算法，保证接近非高斯分布的信息论极限。以往的算法均不能达到优于ε1/2的误差。作为推论，我们在稳健协方差估计上也得到了类似的结果。这两个结果都是基于一个统一的技术。受Diakonikolas等人在鲁棒统计中的最新算法的启发，我们设计了一种基于平方和方法的SDP，用于以下设置:给定X1，…，Xn∈∈k(大k)， n = poly(k)，并承诺从具有有界矩的概率分布中采样X1，…，Xn的子集，恢复该分布的一些信息。
{"title":"Mixture models, robustness, and sum of squares proofs","authors":"Samuel B. Hopkins, Jerry Li","doi":"10.1145/3188745.3188748","DOIUrl":"https://doi.org/10.1145/3188745.3188748","url":null,"abstract":"We use the Sum of Squares method to develop new efficient algorithms for learning well-separated mixtures of Gaussians and robust mean estimation, both in high dimensions, that substantially improve upon the statistical guarantees achieved by previous efficient algorithms. Our contributions are: Mixture models with separated means: We study mixtures of poly(k)-many k-dimensional distributions where the means of every pair of distributions are separated by at least kε. In the special case of spherical Gaussian mixtures, we give a kO(1/ε)-time algorithm that learns the means assuming separation at least kε, for any ε> 0. This is the first algorithm to improve on greedy (“single-linkage”) and spectral clustering, breaking a long-standing barrier for efficient algorithms at separation k1/4. Robust estimation: When an unknown (1−ε)-fraction of X1,…,Xn are chosen from a sub-Gaussian distribution with mean µ but the remaining points are chosen adversarially, we give an algorithm recovering µ to error ε1−1/t in time kO(t), so long as sub-Gaussian-ness up to O(t) moments can be certified by a Sum of Squares proof. This is the first polynomial-time algorithm with guarantees approaching the information-theoretic limit for non-Gaussian distributions. Previous algorithms could not achieve error better than ε1/2. As a corollary, we achieve similar results for robust covariance estimation. Both of these results are based on a unified technique. Inspired by recent algorithms of Diakonikolas et al. in robust statistics, we devise an SDP based on the Sum of Squares method for the following setting: given X1,…,Xn ∈ ℝk for large k and n = poly(k) with the promise that a subset of X1,…,Xn were sampled from a probability distribution with bounded moments, recover some information about that distribution.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":"108 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81633100","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}
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