We study the computational complexity of short sentences in Presburger arithmetic (SHORT-PA). Here by short we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integer coefficients involved in the linear inequalities. We prove that satisfiability of SHORT-PA sentences with m+2 alternating quantifiers is SigmaP_m-complete or PiP_m-complete, when the first quantifier is exists or forall, respectively. Counting versions and restricted systems are also analyzed.
{"title":"Short Presburger Arithmetic Is Hard","authors":"Danny Nguyen, I. Pak","doi":"10.1109/FOCS.2017.13","DOIUrl":"https://doi.org/10.1109/FOCS.2017.13","url":null,"abstract":"We study the computational complexity of short sentences in Presburger arithmetic (SHORT-PA). Here by short we mean sentences with a bounded number of variables, quantifiers, inequalities and Boolean operations; the input consists only of the integer coefficients involved in the linear inequalities. We prove that satisfiability of SHORT-PA sentences with m+2 alternating quantifiers is SigmaP_m-complete or PiP_m-complete, when the first quantifier is exists or forall, respectively. Counting versions and restricted systems are also analyzed.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"124 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122631803","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 problem of computing the Fourier Transform of a signal whose spectrum is dominated by a small number k of frequencies quickly and using a small number of samples of the signal in time domain (the Sparse FFT problem) has received significant attention recently. It is known how to approximately compute the k-sparse Fourier transform in approx klog^2 n time [Hassanieh et alSTOC12], or using the optimal number O(klog n) of samples [Indyk et alFOCS14] in time domain, or come within (loglog n)^{O(1)} factors of both these bounds simultaneously, but no algorithm achieving the optimal O(klog n) bound in sublinear time is known.At a high level, sublinear time Sparse FFT algorithms operate by hashing the spectrum of the input signal into approx k buckets, identifying frequencies that are isolated in their buckets, subtracting them from the signal and repeating until the entire signal is recovered. The notion of isolation in a bucket, inspired by applications of hashing in sparse recovery with arbitrary linear measurements, has been the main tool in the analysis of Fourier hashing schemes in the literature. However, Fourier hashing schemes, which are implemented via filtering, tend to be noisy in the sense that a frequency that hashes into a bucket contributes a non-negligible amount to neighboring buckets. This leakage to neighboring buckets makes identification and estimation challenging, and the standard analysis based on isolation becomes difficult to use without losing Ω(1) factors in sample complexity.In this paper we propose a new technique for analysing noisy hashing schemes that arise in Sparse FFT, which we refer to as isolation on average}. We apply this technique to two problems in Sparse FFT: estimating the values of a list of frequencies using few samples and computing Sparse FFT itself, achieving sample-optimal results in klog^{O(1)} n time for both. We feel that our approach will likely be of interest in designing Fourier sampling schemes for more general settings (e.g. model based Sparse FFT).
在时域中使用少量的信号样本快速计算由少量k个频率占主导的信号的傅里叶变换(稀疏FFT问题)是近年来备受关注的问题。已知如何在approx k log ^ 2n时间内近似计算k-稀疏傅里叶变换[Hassanieh et alSTOC12]，或在时域内使用样本的最优数O(k log n) [Indyk et alFOCS14]，或同时在这两个界的(loglog n{)^O(1)}个因子内，但没有已知的算法在亚线性时间内实现最优O(k log n)界。在高层次上，亚线性时间稀疏FFT算法通过将输入信号的频谱散列到approx k个桶中，识别在其桶中隔离的频率，从信号中减去它们并重复，直到整个信号被恢复。桶中隔离的概念，受到散列在任意线性测量的稀疏恢复中的应用的启发，已经成为文献中分析傅里叶散列方案的主要工具。然而，通过滤波实现的傅里叶哈希方案往往是有噪声的，因为哈希到一个桶中的频率对相邻桶的贡献不可忽略。这种对相邻桶的泄漏使得识别和估计具有挑战性，并且在不丢失样本复杂性因素的情况下，基于隔离的标准分析变得难以使用。在本文中，我们提出了一种新的技术来分析稀疏FFT中出现的噪声哈希方案，我们称之为平均隔离。我们将这种技术应用于稀疏FFT中的两个问题:使用少量样本估计频率列表的值和计算稀疏FFT本身，在k log ^{O(1)} n时间内实现样本最优结果。我们觉得我们的方法可能会对设计更一般设置的傅里叶采样方案感兴趣(例如，基于模型的稀疏FFT)。
{"title":"Sample Efficient Estimation and Recovery in Sparse FFT via Isolation on Average","authors":"M. Kapralov","doi":"10.1109/FOCS.2017.66","DOIUrl":"https://doi.org/10.1109/FOCS.2017.66","url":null,"abstract":"The problem of computing the Fourier Transform of a signal whose spectrum is dominated by a small number k of frequencies quickly and using a small number of samples of the signal in time domain (the Sparse FFT problem) has received significant attention recently. It is known how to approximately compute the k-sparse Fourier transform in approx klog^2 n time [Hassanieh et alSTOC12], or using the optimal number O(klog n) of samples [Indyk et alFOCS14] in time domain, or come within (loglog n)^{O(1)} factors of both these bounds simultaneously, but no algorithm achieving the optimal O(klog n) bound in sublinear time is known.At a high level, sublinear time Sparse FFT algorithms operate by hashing the spectrum of the input signal into approx k buckets, identifying frequencies that are isolated in their buckets, subtracting them from the signal and repeating until the entire signal is recovered. The notion of isolation in a bucket, inspired by applications of hashing in sparse recovery with arbitrary linear measurements, has been the main tool in the analysis of Fourier hashing schemes in the literature. However, Fourier hashing schemes, which are implemented via filtering, tend to be noisy in the sense that a frequency that hashes into a bucket contributes a non-negligible amount to neighboring buckets. This leakage to neighboring buckets makes identification and estimation challenging, and the standard analysis based on isolation becomes difficult to use without losing Ω(1) factors in sample complexity.In this paper we propose a new technique for analysing noisy hashing schemes that arise in Sparse FFT, which we refer to as isolation on average}. We apply this technique to two problems in Sparse FFT: estimating the values of a list of frequencies using few samples and computing Sparse FFT itself, achieving sample-optimal results in klog^{O(1)} n time for both. We feel that our approach will likely be of interest in designing Fourier sampling schemes for more general settings (e.g. model based Sparse FFT).","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128433749","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}
Jack Murtagh, Omer Reingold, Aaron Sidford, S. Vadhan
We give a deterministic tilde{O}(log n)-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and escape probabilities for undirected graphs. Previously, such systems were known to be solvable by randomized algorithms using O(log n) space (Doron, Le Gall, and Ta-Shma, 2017) and hence by deterministic algorithms using O(log^{3/2} n) space (Saks and Zhou, FOCS 1995 and JCSS 1999).Our algorithm combines ideas from time-efficient Laplacian solvers (Spielman and Teng, STOC 04; Peng and Spielman, STOC 14) with ideas used to show that Undirected S-T Connectivity is in deterministic logspace (Reingold, STOC 05 and JACM 08; Rozenman and Vadhan, RANDOM 05).
我们给出了一个确定性的tilde{O} (log n)空间算法，用于近似求解由无向图的拉普拉斯算子给出的线性系统，从而也近似于无向图的命中时间、通勤时间和逃逸概率。此前，已知此类系统可通过使用O(log n)空间的随机算法求解(Doron, Le Gall和Ta-Shma, 2017)，因此可通过使用O(log ^{3/} 2n)空间的确定性算法求解(Saks和Zhou, FOCS 1995和JCSS 1999)。我们的算法结合了时间效率的拉普拉斯解算器的思想(Spielman和Teng, STOC 04;Peng和Spielman, STOC 14)，其思想用于表明无向S-T连接是在确定性对数空间(Reingold, STOC 05和JACM 08;Rozenman and Vadhan, RANDOM 05)。
{"title":"Derandomization Beyond Connectivity: Undirected Laplacian Systems in Nearly Logarithmic Space","authors":"Jack Murtagh, Omer Reingold, Aaron Sidford, S. Vadhan","doi":"10.1109/FOCS.2017.79","DOIUrl":"https://doi.org/10.1109/FOCS.2017.79","url":null,"abstract":"We give a deterministic tilde{O}(log n)-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and escape probabilities for undirected graphs. Previously, such systems were known to be solvable by randomized algorithms using O(log n) space (Doron, Le Gall, and Ta-Shma, 2017) and hence by deterministic algorithms using O(log^{3/2} n) space (Saks and Zhou, FOCS 1995 and JCSS 1999).Our algorithm combines ideas from time-efficient Laplacian solvers (Spielman and Teng, STOC 04; Peng and Spielman, STOC 14) with ideas used to show that Undirected S-T Connectivity is in deterministic logspace (Reingold, STOC 05 and JACM 08; Rozenman and Vadhan, RANDOM 05).","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126381659","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}
A classical difficult isomorphism testing problem is to test isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order. It is known that this problem can be reduced to solving the alternating matrix space isometry problem over a finite field in time polynomial in the underlying vector space size. We propose a venue of attack for the latter problem by viewing it as a linear algebraic analogue of the graph isomorphism problem. This viewpointleads us to explore the possibility of transferring techniques for graph isomorphism to this long-believed bottleneck case of group isomorphism.In 1970s, Babai, Erdős, and Selkow presented the first average-case efficient graph isomorphism testing algorithm (SIAM J Computing, 1980). Inspired by that algorithm, we devise an average-case efficient algorithm for the alternating matrix space isometry problem over a key range of parameters, in a random model of alternating matrix spaces in vein of the Erd∝os-R´enyi model of random graphs. For this, we develop a linear algebraic analogue of the classical individualisation technique, a technique belonging to a set of combinatorial techniques that has been critical for the progress on the worst-case time complexity for graph isomorphism, but was missing in the group isomorphism context. This algorithm also enables us to improve Higmans 57-year-old lower bound on the number of p-groups (Proc. of the LMS, 1960). We finally show that Luks dynamic programming technique for graph isomorphism (STOC 1999) can be adapted to slightly improve the worst-case time complexity of the alternating matrix space isometry problem in a certain range of parameters.Most notable progress on the worst-case time complexity of graph isomorphism, including Babais recent breakthrough (STOC 2016) and Babai and Luks previous record (STOC 1983), has relied on both group theoretic and combinatorial techniques. By developing a linear algebraic analogue of the individualisation technique and demonstrating its usefulness in the average-case setting, the main result opens up the possibility of adapting that strategy for graph isomorphism to this hard instance of group isomorphism. The linear algebraic Erdős-Rényi model is of independent interest and may deserve further study.
一个经典的同构检验难题是在群阶的时间多项式上检验2类p群和指数p群的同构。已知该问题可简化为求解有限域上的交替矩阵空间等距问题，在时间多项式下的基本向量空间大小。我们将后一个问题视为图同构问题的线性代数模拟，提出了一个攻击地点。这种观点引导我们探索将图同构技术转移到长期以来被认为是群同构瓶颈的情况下的可能性。在20世纪70年代，Babai, Erdős和Selkow提出了第一个平均情况下有效的图同构测试算法(SIAM J Computing, 1980)。受该算法的启发，我们在随机图的Erd∝os-R´enyi模型的随机交替矩阵空间模型中，设计了一种针对关键参数范围内的交替矩阵空间等距问题的平均情况有效算法。为此，我们开发了经典个体化技术的线性代数模拟，这种技术属于一组组合技术，对图同构的最坏情况时间复杂度的研究至关重要，但在群同构环境中却缺失了。该算法还使我们能够改进Higmans关于p群数量的57岁下界(Proc. of the LMS, 1960)。我们最后证明了Luks的图同构动态规划技术(STOC 1999)可以在一定参数范围内略微提高交替矩阵空间等距问题的最坏情况时间复杂度。在图同构的最坏情况时间复杂度方面最显著的进展，包括Babai和Luks最近的突破(STOC 2016)和Babai和Luks之前的记录(STOC 1983)，都依赖于群论和组合技术。通过发展个体化技术的线性代数模拟并证明其在平均情况下的有用性，主要结果开辟了将图同构策略适应于群同构的这种困难实例的可能性。线性代数Erdős-Rényi模型具有独立的研究价值，值得进一步研究。
{"title":"Linear Algebraic Analogues of the Graph Isomorphism Problem and the Erdős-Rényi Model","authors":"Yinan Li, Youming Qiao","doi":"10.1109/FOCS.2017.49","DOIUrl":"https://doi.org/10.1109/FOCS.2017.49","url":null,"abstract":"A classical difficult isomorphism testing problem is to test isomorphism of p-groups of class 2 and exponent p in time polynomial in the group order. It is known that this problem can be reduced to solving the alternating matrix space isometry problem over a finite field in time polynomial in the underlying vector space size. We propose a venue of attack for the latter problem by viewing it as a linear algebraic analogue of the graph isomorphism problem. This viewpointleads us to explore the possibility of transferring techniques for graph isomorphism to this long-believed bottleneck case of group isomorphism.In 1970s, Babai, Erdős, and Selkow presented the first average-case efficient graph isomorphism testing algorithm (SIAM J Computing, 1980). Inspired by that algorithm, we devise an average-case efficient algorithm for the alternating matrix space isometry problem over a key range of parameters, in a random model of alternating matrix spaces in vein of the Erd∝os-R´enyi model of random graphs. For this, we develop a linear algebraic analogue of the classical individualisation technique, a technique belonging to a set of combinatorial techniques that has been critical for the progress on the worst-case time complexity for graph isomorphism, but was missing in the group isomorphism context. This algorithm also enables us to improve Higmans 57-year-old lower bound on the number of p-groups (Proc. of the LMS, 1960). We finally show that Luks dynamic programming technique for graph isomorphism (STOC 1999) can be adapted to slightly improve the worst-case time complexity of the alternating matrix space isometry problem in a certain range of parameters.Most notable progress on the worst-case time complexity of graph isomorphism, including Babais recent breakthrough (STOC 2016) and Babai and Luks previous record (STOC 1983), has relied on both group theoretic and combinatorial techniques. By developing a linear algebraic analogue of the individualisation technique and demonstrating its usefulness in the average-case setting, the main result opens up the possibility of adapting that strategy for graph isomorphism to this hard instance of group isomorphism. The linear algebraic Erdős-Rényi model is of independent interest and may deserve further study.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123353335","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 work, we revisit the problem of uniformity testing of discrete probability distributions. A fundamental problem in distribution testing, testing uniformity over a known domain has been addressed over a significant line of works, and is by now fully understood. The complexity of deciding whether an unknown distribution is uniform over its unknown (and arbitrary) support, however, is much less clear. Yet, this task arises as soon as no prior knowledge on the domain is available, or whenever the samples originate from an unknown and unstructured universe.In this work, we introduce and study this generalized uniformity testing question, and establish nearly tight upper and lower bound showing that – quite surprisingly – its sample complexity significantly differs from the known-domain case. Moreover, our algorithm is intrinsically adaptive, in contrast to the overwhelming majority of known distribution testing algorithms.
{"title":"Generalized Uniformity Testing","authors":"Tugkan Batu, C. Canonne","doi":"10.1109/FOCS.2017.86","DOIUrl":"https://doi.org/10.1109/FOCS.2017.86","url":null,"abstract":"In this work, we revisit the problem of uniformity testing of discrete probability distributions. A fundamental problem in distribution testing, testing uniformity over a known domain has been addressed over a significant line of works, and is by now fully understood. The complexity of deciding whether an unknown distribution is uniform over its unknown (and arbitrary) support, however, is much less clear. Yet, this task arises as soon as no prior knowledge on the domain is available, or whenever the samples originate from an unknown and unstructured universe.In this work, we introduce and study this generalized uniformity testing question, and establish nearly tight upper and lower bound showing that – quite surprisingly – its sample complexity significantly differs from the known-domain case. Moreover, our algorithm is intrinsically adaptive, in contrast to the overwhelming majority of known distribution testing algorithms.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"65 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132954349","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 the minimum planarization} problem, given some n-vertex graph, the goal is to find a set of vertices of minimum cardinality whose removal leaves a planar graph. This is a fundamental problem in topological graph theory. We present a log^{O(1)} n-approximation algorithm for this problem on general graphs with running time n^{O(log n/loglog n)}. We also obtain a O(n^≥)-approximation with running time n^{O(1/≥)} for any arbitrarily small constant ≥ 0. Prior to our work, no non-trivial algorithm was known for this problem on general graphs, and the best known result even on graphs of bounded degree was a n^{Ω(1)}-approximation cite{chekuri2013approximation}.As an immediate corollary, we also obtain improved approximation algorithms for the crossing number problem on graphs of bounded degree. Specifically, we obtain O(n^{1/2+≥})-approximation and n^{1/2} log^{O(1)} n-approximation algorithms in time n^{O(1/≥)} and n^{O(log n/loglog n)} respectively. The previously best-known result was a polynomial-time n^{9/10}log^{O(1)} n-approximation algorithm cite{DBLP:conf/stoc/Chuzhoy11}.Our algorithm introduces several new tools including an efficient grid-minor construction for apex graphs, and a new method for computing irrelevant vertices. Analogues of these tools were previously available only for exact algorithms. Our work gives efficient implementations of these ideas in the setting of approximation algorithms, which could be of independent interest.
{"title":"Polylogarithmic Approximation for Minimum Planarization (Almost)","authors":"K. Kawarabayashi, Anastasios Sidiropoulos","doi":"10.1109/FOCS.2017.77","DOIUrl":"https://doi.org/10.1109/FOCS.2017.77","url":null,"abstract":"In the minimum planarization} problem, given some n-vertex graph, the goal is to find a set of vertices of minimum cardinality whose removal leaves a planar graph. This is a fundamental problem in topological graph theory. We present a log^{O(1)} n-approximation algorithm for this problem on general graphs with running time n^{O(log n/loglog n)}. We also obtain a O(n^≥)-approximation with running time n^{O(1/≥)} for any arbitrarily small constant ≥ 0. Prior to our work, no non-trivial algorithm was known for this problem on general graphs, and the best known result even on graphs of bounded degree was a n^{Ω(1)}-approximation cite{chekuri2013approximation}.As an immediate corollary, we also obtain improved approximation algorithms for the crossing number problem on graphs of bounded degree. Specifically, we obtain O(n^{1/2+≥})-approximation and n^{1/2} log^{O(1)} n-approximation algorithms in time n^{O(1/≥)} and n^{O(log n/loglog n)} respectively. The previously best-known result was a polynomial-time n^{9/10}log^{O(1)} n-approximation algorithm cite{DBLP:conf/stoc/Chuzhoy11}.Our algorithm introduces several new tools including an efficient grid-minor construction for apex graphs, and a new method for computing irrelevant vertices. Analogues of these tools were previously available only for exact algorithms. Our work gives efficient implementations of these ideas in the setting of approximation algorithms, which could be of independent interest.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"532 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132312423","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}
Parinya Chalermsook, Marek Cygan, G. Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, L. Trevisan
We consider questions that arise from the intersection between theareas of approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Marx, 2008; Fellow et al., 2012; Downey & Fellow 2013]) are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting opt be the optimum and N be the size of the input, is there an algorithm that runs int(opt) poly(N) time and outputs a solution of size f(opt), forany functions t and f that are independent of N (for Clique, we want f(opt)=Ω(1))? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no o(opt)-FPT-approximation algorithm for Clique and no f(opt)-FPT-approximation algorithm for DomSet, for any function f (e.g., this holds even if f is an exponential or the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur, 2016, Manurangsi & Raghavendra 2016], which states that no 2^{o(n)}-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even (1 - c)-satisfiable for some constant c ≈ 0.Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, the problem of finding maximum subgraphs with hereditary properties (e.g., Maximum Induced Planar Subgraph), and Maximum Induced Matching in bipartite graphs. Previously only exact versions of these problems were known to be W[1]-hard [Lin, 2015; Khot & Raman, 2000; Moser & Sikdar, 2009]. Additionally, we rule out k^{o(1)}-FPT-approximation algorithm for Densest k-Subgraph although this ratio does not yet match the trivial O(k)-approximation algorithm.To the best of our knowledge, prior results only rule out constantfactor approximation for Clique [Hajiaghayi et al., 2013; KK13, Bonnet et al., 2015] and log^{1/4+c}(opt) approximation for DomSet for any constant c ≈ 0 [Chen & Lin, 2016]. Our result on Clique significantly improves on [Hajiaghayi et al., 2013; Bonnet et al., 2015]. However, our result on DomSet is incomparable to [Chen & Lin, 2016] since their results hold under ETH while our results hold under Gap-ETH, which is a stronger assumption.
{"title":"From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More","authors":"Parinya Chalermsook, Marek Cygan, G. Kortsarz, Bundit Laekhanukit, Pasin Manurangsi, Danupon Nanongkai, L. Trevisan","doi":"10.1109/FOCS.2017.74","DOIUrl":"https://doi.org/10.1109/FOCS.2017.74","url":null,"abstract":"We consider questions that arise from the intersection between theareas of approximation algorithms, subexponential-time algorithms, and fixed-parameter tractable algorithms. The questions, which have been asked several times (e.g., [Marx, 2008; Fellow et al., 2012; Downey & Fellow 2013]) are whether there is a non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and Minimum Dominating Set (DomSet) problems parameterized by the size of the optimal solution. In particular, letting opt be the optimum and N be the size of the input, is there an algorithm that runs int(opt) poly(N) time and outputs a solution of size f(opt), forany functions t and f that are independent of N (for Clique, we want f(opt)=Ω(1))? In this paper, we show that both Clique and DomSet admit no non-trivial FPT-approximation algorithm, i.e., there is no o(opt)-FPT-approximation algorithm for Clique and no f(opt)-FPT-approximation algorithm for DomSet, for any function f (e.g., this holds even if f is an exponential or the Ackermann function). In fact, our results imply something even stronger: The best way to solve Clique and DomSet, even approximately, is to essentially enumerate all possibilities. Our results hold under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur, 2016, Manurangsi & Raghavendra 2016], which states that no 2^{o(n)}-time algorithm can distinguish between a satisfiable 3SAT formula and one which is not even (1 - c)-satisfiable for some constant c ≈ 0.Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for Maximum Balanced Biclique, the problem of finding maximum subgraphs with hereditary properties (e.g., Maximum Induced Planar Subgraph), and Maximum Induced Matching in bipartite graphs. Previously only exact versions of these problems were known to be W[1]-hard [Lin, 2015; Khot & Raman, 2000; Moser & Sikdar, 2009]. Additionally, we rule out k^{o(1)}-FPT-approximation algorithm for Densest k-Subgraph although this ratio does not yet match the trivial O(k)-approximation algorithm.To the best of our knowledge, prior results only rule out constantfactor approximation for Clique [Hajiaghayi et al., 2013; KK13, Bonnet et al., 2015] and log^{1/4+c}(opt) approximation for DomSet for any constant c ≈ 0 [Chen & Lin, 2016]. Our result on Clique significantly improves on [Hajiaghayi et al., 2013; Bonnet et al., 2015]. However, our result on DomSet is incomparable to [Chen & Lin, 2016] since their results hold under ETH while our results hold under Gap-ETH, which is a stronger assumption.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"269 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116066788","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 study computing all-pairs shortest paths (APSP) on distributed networks (the CONGEST model). The goal is for every node in the (weighted) network to know the distance from every other node using communication. The problem admits (1+o(1))-approximation Õ(n)-time algorithms [2], [3], which are matched with tilde Ω(n)-time lower bounds [4], [5],footnote{tilde Theta, Õ and tilde Ω hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios.}. No Ω(n) lower bound or o(m) upper bound were known for exact computation.In this paper, we present an Õ(n^{5/4})-time randomized (Las Vegas) algorithm for exact weighted APSP; this provides the first improvement over the naive O(m)-time algorithm when the network is not so sparse. Our result also holds for the case where edge weights are asymmetric} (a.k.a. the directed case where communication is bidirectional). Our techniques also yield an Õ(n^{3/4}k^{1/2}+n)-time algorithm for the k-source shortest paths} problem where we want every node to know distances from k sources; this improves Elkins recent bound [6] when k=tilde Ω(n^{1/4}).We achieve the above results by developing distributed algorithms on top of the classic scaling technique, which we believe is used for the first time for distributed shortest paths computation. One new algorithm which might be of an independent interest is for the reversed r-sink shortest paths} problem, where we want every of r sinks to know its distances from all other nodes, given that every node already knows its distance to every sink. We show an Õ(n√{r})-time algorithm for this problem. Another new algorithm is called short range extension, where we show that in Õ(n√{h}) time the knowledge about distances can be extended for additional h hops. For this, we use weight rounding to introduce small additive} errors which can be later fixed.
{"title":"Distributed Exact Weighted All-Pairs Shortest Paths in Õ(n^{5/4}) Rounds","authors":"Chien-Chung Huang, Danupon Nanongkai, Thatchaphol Saranurak","doi":"10.1109/FOCS.2017.24","DOIUrl":"https://doi.org/10.1109/FOCS.2017.24","url":null,"abstract":"We study computing all-pairs shortest paths (APSP) on distributed networks (the CONGEST model). The goal is for every node in the (weighted) network to know the distance from every other node using communication. The problem admits (1+o(1))-approximation Õ(n)-time algorithms [2], [3], which are matched with tilde Ω(n)-time lower bounds [4], [5],footnote{tilde Theta, Õ and tilde Ω hide polylogarithmic factors. Note that the lower bounds also hold even in the unweighted case and in the weighted case with polynomial approximation ratios.}. No Ω(n) lower bound or o(m) upper bound were known for exact computation.In this paper, we present an Õ(n^{5/4})-time randomized (Las Vegas) algorithm for exact weighted APSP; this provides the first improvement over the naive O(m)-time algorithm when the network is not so sparse. Our result also holds for the case where edge weights are asymmetric} (a.k.a. the directed case where communication is bidirectional). Our techniques also yield an Õ(n^{3/4}k^{1/2}+n)-time algorithm for the k-source shortest paths} problem where we want every node to know distances from k sources; this improves Elkins recent bound [6] when k=tilde Ω(n^{1/4}).We achieve the above results by developing distributed algorithms on top of the classic scaling technique, which we believe is used for the first time for distributed shortest paths computation. One new algorithm which might be of an independent interest is for the reversed r-sink shortest paths} problem, where we want every of r sinks to know its distances from all other nodes, given that every node already knows its distance to every sink. We show an Õ(n√{r})-time algorithm for this problem. Another new algorithm is called short range extension, where we show that in Õ(n√{h}) time the knowledge about distances can be extended for additional h hops. For this, we use weight rounding to introduce small additive} errors which can be later fixed.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"80 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132191921","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}
Danupon Nanongkai, Thatchaphol Saranurak, Christian Wulff-Nilsen
We present a Las Vegas algorithm for dynamically maintaining a minimum spanning forest of an n-node graph undergoing edge insertions and deletions. Our algorithm guarantees an O(n^{o(1)})} worst-case} update time with high probability. This significantly improves the two recent Las Vegas algorithms by Wulff-Nilsen cite{Wulff-Nilsen16a} with update time O(n^{0.5-≥ilon}) for some constant ≥ilon 0 and, independently, by Nanongkai and Saranurak cite{NanongkaiS16} with update time O(n^{0.494}) (the latter works only for maintaining a spanning forest).Our result is obtained by identifying the common framework that both two previous algorithms rely on, and then improve and combine the ideas from both works. There are two main algorithmic components of the framework that are newly improved and critical for obtaining our result. First, we improve the update time from O(n^{0.5-≥ilon}) in cite{Wulff-Nilsen16a} to O(n^{o(1)}) for decrementally removing all low-conductance cuts in an expander undergoing edge deletions. Second, by revisiting the contraction technique by Henzinger and King cite{HenzingerK97b} and Holm et al. cite{HolmLT01, we show a new approach for maintaining a minimum spanning forest in connected graphs with very few (at most (1+o(1))n) edges. This significantly improves the previous approach in cite{Wulff-Nilsen16a, NanongkaiS16} which is based on Fredericksons 2-dimensional topology tree cite{Frederickson85} and illustrates a new application to this old technique.
{"title":"Dynamic Minimum Spanning Forest with Subpolynomial Worst-Case Update Time","authors":"Danupon Nanongkai, Thatchaphol Saranurak, Christian Wulff-Nilsen","doi":"10.1109/FOCS.2017.92","DOIUrl":"https://doi.org/10.1109/FOCS.2017.92","url":null,"abstract":"We present a Las Vegas algorithm for dynamically maintaining a minimum spanning forest of an n-node graph undergoing edge insertions and deletions. Our algorithm guarantees an O(n^{o(1)})} worst-case} update time with high probability. This significantly improves the two recent Las Vegas algorithms by Wulff-Nilsen cite{Wulff-Nilsen16a} with update time O(n^{0.5-≥ilon}) for some constant ≥ilon 0 and, independently, by Nanongkai and Saranurak cite{NanongkaiS16} with update time O(n^{0.494}) (the latter works only for maintaining a spanning forest).Our result is obtained by identifying the common framework that both two previous algorithms rely on, and then improve and combine the ideas from both works. There are two main algorithmic components of the framework that are newly improved and critical for obtaining our result. First, we improve the update time from O(n^{0.5-≥ilon}) in cite{Wulff-Nilsen16a} to O(n^{o(1)}) for decrementally removing all low-conductance cuts in an expander undergoing edge deletions. Second, by revisiting the contraction technique by Henzinger and King cite{HenzingerK97b} and Holm et al. cite{HolmLT01, we show a new approach for maintaining a minimum spanning forest in connected graphs with very few (at most (1+o(1))n) edges. This significantly improves the previous approach in cite{Wulff-Nilsen16a, NanongkaiS16} which is based on Fredericksons 2-dimensional topology tree cite{Frederickson85} and illustrates a new application to this old technique.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131092708","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 consider the problem of robust polynomial regression, where one receives samples that are usually within a small additive error of a target polynomial, but have a chance of being arbitrary adversarial outliers. Previously, it was known how to efficiently estimate the target polynomial only when the outlier probability was subconstant in the degree of the target polynomial. We give an algorithm that works for the entire feasible range of outlier probabilities, while simultaneously improving other parameters of the problem. We complement our algorithm, which gives a factor 2 approximation, with impossibility results that show, for example, that a 1.09 approximation is impossible even with infinitely many samples.
{"title":"Robust Polynomial Regression up to the Information Theoretic Limit","authors":"D. Kane, Sushrut Karmalkar, Eric Price","doi":"10.1109/FOCS.2017.43","DOIUrl":"https://doi.org/10.1109/FOCS.2017.43","url":null,"abstract":"We consider the problem of robust polynomial regression, where one receives samples that are usually within a small additive error of a target polynomial, but have a chance of being arbitrary adversarial outliers. Previously, it was known how to efficiently estimate the target polynomial only when the outlier probability was subconstant in the degree of the target polynomial. We give an algorithm that works for the entire feasible range of outlier probabilities, while simultaneously improving other parameters of the problem. We complement our algorithm, which gives a factor 2 approximation, with impossibility results that show, for example, that a 1.09 approximation is impossible even with infinitely many samples.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126189357","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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