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Proceedings of the forty-seventh annual ACM symposium on Theory of Computing最新文献

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Boolean Function Monotonicity Testing Requires (Almost) n 1/2 Non-adaptive Queries 布尔函数单调性测试需要(几乎)n 1/2非自适应查询
Pub Date : 2014-12-17 DOI: 10.1145/2746539.2746570
Xi Chen, Anindya De, R. Servedio, Li-Yang Tan
We prove a lower bound of Ω(n1/2-c), for all c> 0, on the query complexity of (two-sided error) non-adaptive algorithms for testing whether an n-variable Boolean function is monotone versus constant-far from monotone. This improves a ~Ω(n1/5) lower bound for the same problem that was obtained in [6], and is very close to the recent upper bound of ~O(n1/2/ε2) by Khot et al. [13].
我们证明了(双边误差)非自适应算法的查询复杂度的下界Ω(n1/2-c),对于所有c> 0,用于测试n变量布尔函数是单调还是常数-远离单调。这改进了[6]中得到的相同问题的~Ω(n1/5)下界,并且非常接近Khot等人[13]最近得到的~O(n1/2/ε2)上界。
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引用次数: 53
On the Complexity of Nash Equilibria in Anonymous Games 论匿名博弈中纳什均衡的复杂性
Pub Date : 2014-12-17 DOI: 10.1145/2746539.2746571
X. Chen, D. Durfee, Anthi Orfanou
We show that the problem of finding an ε-approximate Nash equilibrium in an {anonymous} game with seven pure strategies is complete in PPAD, when the approximation parameter ε is exponentially small in the number of players.
我们证明了当近似参数ε在参与人数量上呈指数级小时,在具有7种纯策略的{匿名}对策中寻找ε-近似纳什均衡的问题在PPAD中是完全的。
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引用次数: 24
Beyond the Euler Characteristic: Approximating the Genus of General Graphs 超越欧拉特征:逼近一般图的格
Pub Date : 2014-12-04 DOI: 10.1145/2746539.2746583
K. Kawarabayashi, Anastasios Sidiropoulos
Computing the Euler genus of a graph is a fundamental problem in graph theory and topology. It has been shown to be NP-hard by Thomassen [27] and a linear-time fixed-parameter algorithm has been obtained by Mohar [20]. Despite extensive study, the approximability of the Euler genus remains wide open. While the existence of a constant factor approximation is not ruled out, the currently best-known upper bound is a trivial O(n/g)-approximation that follows from bounds on the Euler characteristic. In this paper, we give the first non-trivial approximation algorithm for this problem. Specifically, we present a polynomial-time algorithm which given a graph G of Euler genus g outputs an embedding of G into a surface of Euler genus gO(1). Combined with the above O(n/g)-approximation, our result also implies a O(n1-α)-approximation, for some universal constant α> 0. Our approximation algorithm also has implications for the design of algorithms on graphs of small genus. Several of these algorithms require that an embedding of the graph into a surface of small genus is given as part of the input. Our result implies that many of these algorithms can be implemented even when the embedding of the input graph is unknown.
图的欧拉格的计算是图论和拓扑学中的一个基本问题。Thomassen[27]证明了它是NP-hard的,Mohar[20]给出了线性时间固定参数算法。尽管有广泛的研究,欧拉属的近似性仍然是开放的。虽然不排除存在常数因子近似值,但目前最著名的上界是一个平凡的O(n/g)近似值,它遵循欧拉特征的边界。本文给出了该问题的第一个非平凡逼近算法。具体来说,我们提出了一种多项式时间算法,该算法给定欧拉属G的图G,输出G嵌入欧拉属gO(1)的曲面。结合上述的O(n/g)-近似,我们的结果也暗示了对于某些普遍常数α> 0的O(n1-α)-近似。我们的近似算法对小属图的算法设计也有启示。其中一些算法要求将图嵌入到一个小属的曲面中作为输入的一部分。我们的结果表明,即使输入图的嵌入是未知的,许多这些算法也可以实现。
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引用次数: 10
Near Optimal LP Rounding Algorithm for CorrelationClustering on Complete and Complete k-partite Graphs 完备和完备k部图相关聚类的近最优LP舍入算法
Pub Date : 2014-12-01 DOI: 10.1145/2746539.2746604
Shuchi Chawla, K. Makarychev, T. Schramm, G. Yaroslavtsev
We give new rounding schemes for the standard linear programming relaxation of the correlation clustering problem, achieving approximation factors almost matching the integrality gaps: For complete graphs our approximation is 2.06 - ε, which almost matches the previously known integrality gap of 2. For complete k-partite graphs our approximation is 3. We also show a matching integrality gap. For complete graphs with edge weights satisfying triangle inequalities and probability constraints, our approximation is 1.5, and we show an integrality gap of 1.2. Our results improve a long line of work on approximation algorithms for correlation clustering in complete graphs, previously culminating in a ratio of 2.5 for the complete case by Ailon, Charikar and Newman (JACM'08). In the weighted complete case satisfying triangle inequalities and probability constraints, the same authors give a 2-approximation; for the bipartite case, Ailon, Avigdor-Elgrabli, Liberty and van Zuylen give a 4-approximation (SICOMP'12).
对于相关聚类问题的标准线性规划松弛,我们给出了新的舍入方案,实现了近似因子几乎与完整性间隙匹配:对于完全图,我们的近似因子为2.06 - ε,几乎与之前已知的2的完整性间隙匹配。对于完全k部图,我们的近似值是3。我们还展示了一个匹配的完整性差距。对于边权满足三角形不等式和概率约束的完全图,我们的近似值为1.5,并且我们显示了1.2的完整性间隙。我们的结果改进了在完全图中相关聚类的近似算法方面的一长串工作,之前由Ailon, Charikar和Newman (JACM'08)在完全情况下达到了2.5的比率。在满足三角形不等式和概率约束的加权完全情况下,同样的作者给出了一个2逼近;对于两部分的情况,Ailon, Avigdor-Elgrabli, Liberty和van Zuylen给出了一个4近似(SICOMP'12)。
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引用次数: 96
Edit Distance Cannot Be Computed in Strongly Subquadratic Time (unless SETH is false) 编辑距离不能在强次二次时间内计算(除非SETH为假)
Pub Date : 2014-11-30 DOI: 10.1145/2746539.2746612
A. Backurs, P. Indyk
The edit distance (a.k.a. the Levenshtein distance) between two strings is defined as the minimum number of insertions, deletions or substitutions of symbols needed to transform one string into another. The problem of computing the edit distance between two strings is a classical computational task, with a well-known algorithm based on dynamic programming. Unfortunately, all known algorithms for this problem run in nearly quadratic time. In this paper we provide evidence that the near-quadratic running time bounds known for the problem of computing edit distance might be {tight}. Specifically, we show that, if the edit distance can be computed in time O(n2-δ) for some constant δ>0, then the satisfiability of conjunctive normal form formulas with N variables and M clauses can be solved in time MO(1) 2(1-ε)N for a constant ε>0. The latter result would violate the Strong Exponential Time Hypothesis, which postulates that such algorithms do not exist.
两个字符串之间的编辑距离(又称Levenshtein距离)定义为将一个字符串转换为另一个字符串所需的符号插入、删除或替换的最小数量。计算两个字符串之间的编辑距离问题是一个经典的计算任务,采用了一种著名的基于动态规划的算法。不幸的是,所有已知的解决这个问题的算法都在接近二次的时间内运行。在本文中,我们提供的证据表明,已知的计算编辑距离问题的近二次运行时间界限可能是{紧}的。具体地说,我们证明了如果对于某个常数δ>0,编辑距离可以在O(n2-δ)时间内计算,那么对于一个常数ε>0,具有N个变量和M个子句的合取范式公式的可满足性可以在MO(1) 2(1-ε)N时间内求解。后一种结果将违反强指数时间假设,该假设假定这种算法不存在。
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引用次数: 379
FPTAS for #BIS with Degree Bounds on One Side 在一侧有度界的#BIS的FPTAS
Pub Date : 2014-11-28 DOI: 10.1145/2746539.2746598
Jingcheng Liu, P. Lu
Counting the number of independent sets for a bipartite graph (#BIS) plays a crucial role in the study of approximate counting. It has been conjectured that there is no fully polynomial-time (randomized) approximation scheme (FPTAS/FPRAS) for #BIS, and it was proved that the problem for instances with a maximum degree of 6 is already as hard as the general problem. In this paper, we obtain a surprising tractability result for a family of #BIS instances. We design a very simple deterministic fully polynomial-time approximation scheme (FPTAS) for #BIS when the maximum degree for one side is no larger than 5. There is no restriction for the degrees on the other side, which do not even have to be bounded by a constant. Previously, FPTAS was only known for instances with a maximum degree of 5 for both sides.
二部图(#BIS)的独立集数的计算在近似计数的研究中起着至关重要的作用。对于#BIS,我们推测不存在完全多项式时间(随机)近似方案(FPTAS/FPRAS),并证明了对于最大次数为6的实例的问题已经和一般问题一样难。在本文中,我们获得了一组#BIS实例的令人惊讶的可追溯性结果。我们为#BIS设计了一个非常简单的确定性全多项式时间近似方案(FPTAS),其中一侧的最大度不大于5。另一边的度数没有限制,甚至不需要以常数为界。以前,自由贸易协定只在双方最高程度为5的情况下才为人所知。
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引用次数: 24
Randomized Rounding for the Largest Simplex Problem 最大单纯形问题的随机舍入
Pub Date : 2014-11-28 DOI: 10.1145/2746539.2746628
Aleksandar Nikolov
The maximum volume j-simplex problem asks to compute the j-dimensional simplex of maximum volume inside the convex hull of a given set of n points in Qd. We give a deterministic approximation algorithm for this problem which achieves an approximation ratio of ej/2 + o(j). The problem is known to be NP-hard to approximate within a factor of cj for some constant c > 1. Our algorithm also gives a factor ej + o(j) approximation for the problem of finding the principal j x j submatrix of a rank d positive semidefinite matrix with the largest determinant. We achieve our approximation by rounding solutions to a generalization of the D-optimal design problem, or, equivalently, the dual of an appropriate smallest enclosing ellipsoid problem. Our arguments give a short and simple proof of a restricted invertibility principle for determinants.
最大体积j-单纯形问题要求计算Qd中给定n个点的凸包内最大体积的j维单纯形。给出了一种确定性逼近算法,逼近比为ej/2 + o(j)。对于某个常数c > 1,这个问题已知是np困难的,难以在系数cj内近似。我们的算法也给出了一个因子ej + o(j)近似的问题,以找到一个最大行列式的第d阶正半定矩阵的主jxj子矩阵。我们通过四舍五入的方法来逼近d -最优设计问题的一般化解,或者,等价地,一个适当的最小封闭椭球问题的对偶。我们的论证给出了行列式的有限可逆性原理的一个简短的证明。
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引用次数: 59
Almost Optimal Pseudorandom Generators for Spherical Caps: Extended Abstract 球面帽的几乎最优伪随机发生器:扩展摘要
Pub Date : 2014-11-23 DOI: 10.1145/2746539.2746611
Pravesh Kothari, R. Meka
Halfspaces or linear threshold functions are widely studied in complexity theory, learning theory and algorithm design. In this work we study the natural problem of constructing pseudorandom generators (PRGs) for halfspaces over the sphere, aka spherical caps, which besides being interesting and basic geometric objects, also arise frequently in the analysis of various randomized algorithms (e.g., randomized rounding). We give an explicit PRG which fools spherical caps within error ε and has an almost optimal seed-length of O(log n + log(1/ε) ⋅ log log(1/ε)). For an inverse-polynomially growing error ε, our generator has a seed-length optimal up to a factor of O( log log (n)). The most efficient PRG previously known (due to Kane 2012) requires a seed-length of Ω(log3/2(n)) in this setting. We also obtain similar constructions to fool halfspaces with respect to the Gaussian distribution. Our construction and analysis are significantly different from previous works on PRGs for halfspaces and build on the iterative dimension reduction ideas of Kane et. al. 2011 and Celis et. al. 2013, the classical moment problem from probability theory and explicit constructions of approximate orthogonal designs based on the seminal work of Bourgain and Gamburd 2011 on expansion in Lie groups.
半空间或线性阈值函数在复杂性理论、学习理论和算法设计中得到了广泛的研究。在这项工作中,我们研究了为球体上的半空间构造伪随机生成器(prg)的自然问题,即球形帽,它除了是有趣和基本的几何对象外,还经常出现在各种随机算法的分析中(例如,随机四舍五入)。我们给出了一个显式PRG,它可以在误差ε范围内欺骗球形帽,并且具有几乎最优的种子长度O(log n + log(1/ε)·log log(1/ε))。对于一个逆多项式增长的误差ε,我们的生成器具有一个最优的种子长度到O(log log (n))的因子。在这种情况下,已知的最有效的PRG(由于Kane 2012)要求种子长度为Ω(log3/2(n))。我们也得到了类似的构造来愚弄相对于高斯分布的半空间。我们的构造和分析与之前关于半空间prg的研究有很大的不同,我们基于Kane等人2011年和Celis等人2013年的迭代降维思想、概率论中的经典矩问题以及基于Bourgain和Gamburd 2011年关于李群展开的开创性工作的近似正交设计的显式构造。
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引用次数: 17
Lower Bounds on the Size of Semidefinite Programming Relaxations 半定规划松弛大小的下界
Pub Date : 2014-11-23 DOI: 10.1145/2746539.2746599
James R. Lee, P. Raghavendra, David Steurer
We introduce a method for proving lower bounds on the efficacy of semidefinite programming (SDP) relaxations for combinatorial problems. In particular, we show that the cut, TSP, and stable set polytopes on n-vertex graphs are not the linear image of the feasible region of any SDP (i.e., any spectrahedron) of dimension less than 2nδ, for some constant δ > 0. This result yields the first super-polynomial lower bounds on the semidefinite extension complexity of any explicit family of polytopes. Our results follow from a general technique for proving lower bounds on the positive semidefinite rank of a matrix. To this end, we establish a close connection between arbitrary SDPs and those arising from the sum-of-squares SDP hierarchy. For approximating maximum constraint satisfaction problems, we prove that SDPs of polynomial-size are equivalent in power to those arising from degree-O(1) sum-of-squares relaxations. This result implies, for instance, that no family of polynomial-size SDP relaxations can achieve better than a 7/8-approximation for max-sat.
介绍了一种证明组合问题半定规划松弛有效性下界的方法。特别地,我们证明了n顶点图上的切多面体、TSP多面体和稳定集多面体不是任何小于2nδ维数的SDP(即任何谱面体)可行域的线性像,对于某些常数δ > 0。这一结果给出了任何显族多面体的半定扩展复杂度的第一个超多项式下界。我们的结果来自于证明矩阵正半定秩下界的一般技术。为此,我们建立了任意SDP和由平方和SDP层次产生的SDP之间的密切联系。对于逼近最大约束满足问题,我们证明了多项式大小的sdp与由o(1)次平方和松弛引起的sdp在幂次上是相等的。这个结果意味着,例如,没有一个多项式大小的SDP松弛族可以获得比max-sat的7/8近似更好的结果。
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引用次数: 181
Efficiently Learning Ising Models on Arbitrary Graphs 有效地学习任意图上的Ising模型
Pub Date : 2014-11-22 DOI: 10.1145/2746539.2746631
Guy Bresler
graph underlying an Ising model from i.i.d. samples. Over the last fifteen years this problem has been of significant interest in the statistics, machine learning, and statistical physics communities, and much of the effort has been directed towards finding algorithms with low computational cost for various restricted classes of models. Nevertheless, for learning Ising models on general graphs with p nodes of degree at most d, it is not known whether or not it is possible to improve upon the pd computation needed to exhaustively search over all possible neighborhoods for each node. In this paper we show that a simple greedy procedure allows to learn the structure of an Ising model on an arbitrary bounded-degree graph in time on the order of p2. We make no assumptions on the parameters except what is necessary for identifiability of the model, and in particular the results hold at low-temperatures as well as for highly non-uniform models. The proof rests on a new structural property of Ising models: we show that for any node there exists at least one neighbor with which it has a high mutual information.
基于iid样本的Ising模型的图。在过去的15年里,这个问题在统计学、机器学习和统计物理社区中引起了极大的兴趣,并且很多努力都是为了寻找各种受限制的模型类的低计算成本算法。然而,对于在p个节点最多为d度的一般图上学习Ising模型,是否有可能改进pd计算,以便对每个节点的所有可能邻域进行穷举搜索,目前尚不清楚。在本文中,我们证明了一个简单的贪心过程可以在时间上以p2阶学习任意有界度图上的Ising模型的结构。除了模型的可识别性所必需的参数外,我们不对参数作任何假设,特别是在低温和高度不均匀的模型下,结果都是成立的。该证明基于Ising模型的一个新的结构性质:我们证明了对于任何节点,存在至少一个与其具有高互信息的邻居。
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引用次数: 178
期刊
Proceedings of the forty-seventh annual ACM symposium on Theory of Computing
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