2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)最新文献

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}

{"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}