We consider the problem of efficiently learning mixtures of a large number of spherical Gaussians, when the components of the mixture are well separated. In the most basic form of this problem, we are given samples from a uniform mixture of k standard spherical Gaussians with means mu_1,...,mu_k in R^d, and the goal is to estimate the means up to accuracy δ using poly(k,d, 1/δ) samples.In this work, we study the following question: what is the minimum separation needed between the means for solving this task? The best known algorithm due to Vempala and Wang [JCSS 2004] requires a separation of roughly min{k,d}^{1/4}. On the other hand, Moitra and Valiant [FOCS 2010] showed that with separation o(1), exponentially many samples are required. We address the significant gap between these two bounds, by showing the following results.• We show that with separation o(√log k), super-polynomially many samples are required. In fact, this holds even when the k means of the Gaussians are picked at random in d=O(log k) dimensions.• We show that with separation Ω(√log k), poly(k,d,1/δ) samples suffice. Notice that the bound on the separation is independent of δ. This result is based on a new and efficient accuracy boosting algorithm that takes as input coarse estimates of the true means and in time (and samples) poly(k,d, 1δ) outputs estimates of the means up to arbitrarily good accuracy δ assuming the separation between the means is Ωmin √(log k),√d) (independently of δ). The idea of the algorithm is to iteratively solve a diagonally dominant system of non-linear equations.We also (1) present a computationally efficient algorithm in d=O(1) dimensions with only Ω(√{d}) separation, and (2) extend our results to the case that components might have different weights and variances. These results together essentially characterize the optimal order of separation between components that is needed to learn a mixture of k spherical Gaussians with polynomial samples.
{"title":"On Learning Mixtures of Well-Separated Gaussians","authors":"O. Regev, Aravindan Vijayaraghavan","doi":"10.1109/FOCS.2017.17","DOIUrl":"https://doi.org/10.1109/FOCS.2017.17","url":null,"abstract":"We consider the problem of efficiently learning mixtures of a large number of spherical Gaussians, when the components of the mixture are well separated. In the most basic form of this problem, we are given samples from a uniform mixture of k standard spherical Gaussians with means mu_1,...,mu_k in R^d, and the goal is to estimate the means up to accuracy δ using poly(k,d, 1/δ) samples.In this work, we study the following question: what is the minimum separation needed between the means for solving this task? The best known algorithm due to Vempala and Wang [JCSS 2004] requires a separation of roughly min{k,d}^{1/4}. On the other hand, Moitra and Valiant [FOCS 2010] showed that with separation o(1), exponentially many samples are required. We address the significant gap between these two bounds, by showing the following results.• We show that with separation o(√log k), super-polynomially many samples are required. In fact, this holds even when the k means of the Gaussians are picked at random in d=O(log k) dimensions.• We show that with separation Ω(√log k), poly(k,d,1/δ) samples suffice. Notice that the bound on the separation is independent of δ. This result is based on a new and efficient accuracy boosting algorithm that takes as input coarse estimates of the true means and in time (and samples) poly(k,d, 1δ) outputs estimates of the means up to arbitrarily good accuracy δ assuming the separation between the means is Ωmin √(log k),√d) (independently of δ). The idea of the algorithm is to iteratively solve a diagonally dominant system of non-linear equations.We also (1) present a computationally efficient algorithm in d=O(1) dimensions with only Ω(√{d}) separation, and (2) extend our results to the case that components might have different weights and variances. These results together essentially characterize the optimal order of separation between components that is needed to learn a mixture of k spherical Gaussians with polynomial samples.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122470015","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}
Non-malleable commitments, introduced by Dolev, Dwork and Naor (STOC 1991), are a fundamental cryptographic primitive, and their round complexity has been a subject of great interest. And yet, the goal of achieving non-malleable commitments with only one or two rounds} has been elusive. Pass (TCC 2013) captured this difficulty by proving important impossibility results regarding two-round non-malleable commitments. This led to the widespread belief that achieving two-round non-malleable commitments was impossible from standard assumptions. We show that this belief was false. Indeed, we obtain the following positive results:∘ We construct two-message non-malleable commitments satisfying non-malleability with respect to commitment, based on standard sub-exponential assumptions, namely: sub-exponential one-way permutations, sub-exponential ZAPs, and sub-exponential DDH. Furthermore, our protocol is public-coin}.∘ We obtain two-message private-coin} non-malleable commitments with respect to commitment, assuming only sub-exponential DDH or QR or N^{th}-residuosity.∘ We bootstrap the above protocols (under the same assumptions) to obtain two round constant bounded-concurrent non-malleable commitments. In the simultaneous message model, we obtain unbounded concurrent non-malleability in two rounds.∘ In the simultaneous messages model, we obtain one-round} non-malleable commitments, with unbounded concurrent security with respect to opening, under standard sub-exponential assumptions.– This implies non-interactive non-malleable commitments with respect to opening, in a restricted model with a broadcast channel, and a-priori bounded polynomially many parties such that every party is aware of every other party in the system. To the best of our knowledge, this is the first protocol to achieve completely non-interactive non-malleability in any plain model setting from standard assumptions.– As an application of this result, in the simultaneous exchange model, we obtain two-round multi-party pseudorandom coin-flipping.∘ We construct two-message zero-knowledge arguments with super-polynomial strong} simulation (SPSS-ZK), which also serve as an important tool for our constructions of non-malleable commitments.∘ In order to obtain our results, we develop several techniques that may be of independent interest.– We give the first two-round black-box rewinding strategy based on standard sub-exponential assumptions, in the plain model.– We also give a two-round tag amplification technique for non-malleable commitments, that amplifies a 4-tag scheme to a scheme for all tags, while relying on sub-exponential DDH. This includes a more efficient alternative to the DDN encoding.The full version of this paper is available online at: https://eprint.iacr.org/2017/291.pdf.
{"title":"How to Achieve Non-Malleability in One or Two Rounds","authors":"Dakshita Khurana, A. Sahai","doi":"10.1109/FOCS.2017.58","DOIUrl":"https://doi.org/10.1109/FOCS.2017.58","url":null,"abstract":"Non-malleable commitments, introduced by Dolev, Dwork and Naor (STOC 1991), are a fundamental cryptographic primitive, and their round complexity has been a subject of great interest. And yet, the goal of achieving non-malleable commitments with only one or two rounds} has been elusive. Pass (TCC 2013) captured this difficulty by proving important impossibility results regarding two-round non-malleable commitments. This led to the widespread belief that achieving two-round non-malleable commitments was impossible from standard assumptions. We show that this belief was false. Indeed, we obtain the following positive results:∘ We construct two-message non-malleable commitments satisfying non-malleability with respect to commitment, based on standard sub-exponential assumptions, namely: sub-exponential one-way permutations, sub-exponential ZAPs, and sub-exponential DDH. Furthermore, our protocol is public-coin}.∘ We obtain two-message private-coin} non-malleable commitments with respect to commitment, assuming only sub-exponential DDH or QR or N^{th}-residuosity.∘ We bootstrap the above protocols (under the same assumptions) to obtain two round constant bounded-concurrent non-malleable commitments. In the simultaneous message model, we obtain unbounded concurrent non-malleability in two rounds.∘ In the simultaneous messages model, we obtain one-round} non-malleable commitments, with unbounded concurrent security with respect to opening, under standard sub-exponential assumptions.– This implies non-interactive non-malleable commitments with respect to opening, in a restricted model with a broadcast channel, and a-priori bounded polynomially many parties such that every party is aware of every other party in the system. To the best of our knowledge, this is the first protocol to achieve completely non-interactive non-malleability in any plain model setting from standard assumptions.– As an application of this result, in the simultaneous exchange model, we obtain two-round multi-party pseudorandom coin-flipping.∘ We construct two-message zero-knowledge arguments with super-polynomial strong} simulation (SPSS-ZK), which also serve as an important tool for our constructions of non-malleable commitments.∘ In order to obtain our results, we develop several techniques that may be of independent interest.– We give the first two-round black-box rewinding strategy based on standard sub-exponential assumptions, in the plain model.– We also give a two-round tag amplification technique for non-malleable commitments, that amplifies a 4-tag scheme to a scheme for all tags, while relying on sub-exponential DDH. This includes a more efficient alternative to the DDN encoding.The full version of this paper is available online at: https://eprint.iacr.org/2017/291.pdf.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"12 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117353469","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 give an adaptive algorithm that tests whether an unknown Boolean function f: {0,1}^n -≈ {0, 1} is unate (i.e. every variable of f is either non-decreasing or non-increasing) or ≥-far from unate with one-sided error and O(n^{3/4}/≥^2) many queries. This improves on the best adaptive O(n/≥)-query algorithm from Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova and Seshadhri when 1/ε
{"title":"Boolean Unateness Testing with Õ(n^{3/4}) Adaptive Queries","authors":"Xi Chen, Erik Waingarten, Jinyu Xie","doi":"10.1109/FOCS.2017.85","DOIUrl":"https://doi.org/10.1109/FOCS.2017.85","url":null,"abstract":"We give an adaptive algorithm that tests whether an unknown Boolean function f: {0,1}^n -≈ {0, 1} is unate (i.e. every variable of f is either non-decreasing or non-increasing) or ≥-far from unate with one-sided error and O(n^{3/4}/≥^2) many queries. This improves on the best adaptive O(n/≥)-query algorithm from Baleshzar, Chakrabarty, Pallavoor, Raskhodnikova and Seshadhri when 1/ε","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"35 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128296105","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 set disjointess problem, we have k players, each with a private input X^i ⊆ [n], and the goal is for the players to determine whether or not their sets have a global intersection. The players communicate over a shared blackboard, and we charge them for each bit that they write on the board.We study the trade-off between the number of interaction rounds we allow the players, and the total number of bits they must send to solve set disjointness. We show that if R rounds of interaction are allowed, the communication cost is Ω(nk^{1/R}/R^4), which is nearly tight. We also leverage our proof to show that wellfare maximization with unit demand bidders cannot be solved efficiently in a small number of rounds: here, we have k players bidding on n items, and the goal is to find a matching between items and player that bid on them which approximately maximizes the total number of items assigned. It was previously shown by Alon et. al. that Ω(log log k) rounds of interaction are required to find an assignment which achieves a constant approximation to the maximum-wellfare assignment, even if each player is allowed to write n^{≥(R)} bits on the board in each round, where ≥(R) = exp(-R). We improve this lower bound to Ωlog k / log log k), which is known to be tight up to a log log k factor.
{"title":"A Rounds vs. Communication Tradeoff for Multi-Party Set Disjointness","authors":"M. Braverman, R. Oshman","doi":"10.1109/FOCS.2017.22","DOIUrl":"https://doi.org/10.1109/FOCS.2017.22","url":null,"abstract":"In the set disjointess problem, we have k players, each with a private input X^i ⊆ [n], and the goal is for the players to determine whether or not their sets have a global intersection. The players communicate over a shared blackboard, and we charge them for each bit that they write on the board.We study the trade-off between the number of interaction rounds we allow the players, and the total number of bits they must send to solve set disjointness. We show that if R rounds of interaction are allowed, the communication cost is Ω(nk^{1/R}/R^4), which is nearly tight. We also leverage our proof to show that wellfare maximization with unit demand bidders cannot be solved efficiently in a small number of rounds: here, we have k players bidding on n items, and the goal is to find a matching between items and player that bid on them which approximately maximizes the total number of items assigned. It was previously shown by Alon et. al. that Ω(log log k) rounds of interaction are required to find an assignment which achieves a constant approximation to the maximum-wellfare assignment, even if each player is allowed to write n^{≥(R)} bits on the board in each round, where ≥(R) = exp(-R). We improve this lower bound to Ωlog k / log log k), which is known to be tight up to a log log k factor.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130322639","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 present a novel input sensitive analysis of a deterministic online algorithm cite{r_approx16} for the minimum metric bipartite matching problem. We show that, in the adversarial model, for any metric space metric and a set of n servers S, the competitive ratio of this algorithm is O(mu_{metric}(S)log^2 n); here mu_{metric}(S) is the maximum ratio of the traveling salesman tour and the diameter of any subset of S. It is straight-forward to show that any algorithm, even with complete knowledge of metric and S, will have a competitive ratio of Ω(mu_metric(S)). So, the performance of this algorithm is sensitive to the input and near-optimal for any given S and metric. As consequences, we also achieve the following results:• If S is a set of points on a line, then mu_metric(S) = Theta(1) and the competitive ratio is O(log^2 n), and,• If S is a set of points spanning a subspace with doubling dimension d, then mu_metric(S) = O(n^{1-1/d}) and the competitive ratio is O(n^{1-1/d}log^2 n).Prior to this result, the previous best-known algorithm for the line metric has a competitive ratio of O(n^{0.59}) and requires both S and the request set R to be on a line. There is also an O(log n) competitive algorithm in the weaker oblivious adversary model.To obtain our results, we partition the requests into well-separated clusters and replace each cluster with a small and a large weighted ball; the weight of a ball is the number of requests in the cluster. We show that the cost of the online matching can be expressed as the sum of the weight times radius of the smaller balls. We also show that the cost of edges of the optimal matching inside each larger ball can be shown to be proportional to the weight times the radius of the larger ball. We then use a simple variant of the well-known Vitalis covering lemma to relate the radii of these balls and obtain the competitive ratio.
针对最小度量二部匹配问题，提出了一种新的确定性在线算法cite{r_approx16}的输入敏感性分析。我们证明，在对抗模型中，对于任意度量空间metric和一组n个服务器S，该算法的竞争比为O(mu ＿ {metric} (S) log ^2 n);这里mu ＿ {metric} (S)是旅行推销员的行程和S的任意子集的直径的最大比值。这很直接地表明，任何算法，即使完全了解metric和S，也会有一个竞争比＆＃x03A9;(mu ＿ metric (S))。因此，该算法的性能对输入很敏感，对于任何给定的S和metric都接近最优。作为结果，我们还实现了以下结果:＆＃x2022;如果S是直线上的点的集合，则mu ＿ metric (S) = Theta(1)，竞争比为O(log ^2 n)， ＆＃x2022;如果S是一个点的集合，它生成了一个具有倍维d的子空间，那么mu ＿ metric (S) = O(n^{1-1/d})，竞争比为O(n^{1-1/d}log ^2 n)。在此结果之前，之前最著名的线度量算法的竞争比为O(n^{0.59})，并且要求S和请求集R都在一条线上。在弱遗忘对手模型中也有一个O(log n)竞争算法。为了获得我们的结果，我们将请求划分为分离良好的簇，并用一个小的和一个大的加权球代替每个簇;球的权重是集群中请求的数量。我们证明了在线匹配的代价可以表示为小球的重量乘以半径的总和。我们还表明，每个大球内最优匹配的边的成本可以显示为与权重乘以大球的半径成正比。然后，我们使用著名的Vitalis覆盖引理的一个简单变体来关联这些球的半径并获得竞争比。
{"title":"An Input Sensitive Online Algorithm for the Metric Bipartite Matching Problem","authors":"K. Nayyar, S. Raghvendra","doi":"10.1109/FOCS.2017.53","DOIUrl":"https://doi.org/10.1109/FOCS.2017.53","url":null,"abstract":"We present a novel input sensitive analysis of a deterministic online algorithm cite{r_approx16} for the minimum metric bipartite matching problem. We show that, in the adversarial model, for any metric space metric and a set of n servers S, the competitive ratio of this algorithm is O(mu_{metric}(S)log^2 n); here mu_{metric}(S) is the maximum ratio of the traveling salesman tour and the diameter of any subset of S. It is straight-forward to show that any algorithm, even with complete knowledge of metric and S, will have a competitive ratio of Ω(mu_metric(S)). So, the performance of this algorithm is sensitive to the input and near-optimal for any given S and metric. As consequences, we also achieve the following results:• If S is a set of points on a line, then mu_metric(S) = Theta(1) and the competitive ratio is O(log^2 n), and,• If S is a set of points spanning a subspace with doubling dimension d, then mu_metric(S) = O(n^{1-1/d}) and the competitive ratio is O(n^{1-1/d}log^2 n).Prior to this result, the previous best-known algorithm for the line metric has a competitive ratio of O(n^{0.59}) and requires both S and the request set R to be on a line. There is also an O(log n) competitive algorithm in the weaker oblivious adversary model.To obtain our results, we partition the requests into well-separated clusters and replace each cluster with a small and a large weighted ball; the weight of a ball is the number of requests in the cluster. We show that the cost of the online matching can be expressed as the sum of the weight times radius of the smaller balls. We also show that the cost of edges of the optimal matching inside each larger ball can be shown to be proportional to the weight times the radius of the larger ball. We then use a simple variant of the well-known Vitalis covering lemma to relate the radii of these balls and obtain the competitive ratio.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126594024","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}
Dynamic programming is a basic, and one of the most systematic techniques for developing polynomial time algorithms with overwhelming applications. However, it often suffers from having high running time and space complexity due to (a) maintaining a table of solutions for a large number of sub-instances, and (b) combining/comparing these solutions to successively solve larger sub-instances. In this paper, we consider a canonical cubic time and quadratic space dynamic programming, and show how improvements in both its time and space uses are possible. As a result, we obtain fast small-space approximation algorithms for the fundamental problems of context free grammar recognition} (the basic computer science problem of parsing), the language edit distance} (a significant generalization of string edit distance and parsing), and RNA folding} (a classical problem in bioinformatics). For these problems, ours are the first algorithms that break the cubic-time barrier of any combinatorial algorithm, and quadratic-space barrier of any algorithm significantly improving upon their long-standing space and time complexities. Our technique applies to many other problems as well including string edit distance computation, and finding longest increasing subsequence.Our improvements come from directly grinding the dynamic programming and looking through the lens of language edit distance which generalizes both context free grammar recognition, and RNA folding. From known conditional lower bound results, neither of these problems can have an exact combinatorial algorithm (one that does not use fast matrix multiplication) running in truly subcubic time. Moreover, for language edit distance such an algorithm cannot exist even when nontrivial multiplicative approximation is allowed. We overcome this hurdle by designing an additive-approximation algorithm that for any parameter k ≈ 0, uses O(nklog{n}) space and O(n^2klog{n}) time and provides an additive O(frac{n}{k}log{n})-approximation. In particular, in tilde{O}(n)footnotemark[1] space and tilde{O}(n^2) time it can solve deterministically whether a string belongs to a context free grammar, or ≥ilon-far from it for any constant ≥ilon ≈ 0. We also improve the above results to obtain an algorithm that outputs an ≥ilon⋅ n-additive approximation to the above problems with space complexity O(n^{2/3} log{n}). The space complexity remains sublinear in n, as long as ≥ilon = o(n^{-frac{1}{4}}). Moreover, we provide the first MapReduce and streaming algorithms for them with multiple passes and sublinear space complexity.
{"title":"Fast & Space-Efficient Approximations of Language Edit Distance and RNA Folding: An Amnesic Dynamic Programming Approach","authors":"B. Saha","doi":"10.1109/FOCS.2017.35","DOIUrl":"https://doi.org/10.1109/FOCS.2017.35","url":null,"abstract":"Dynamic programming is a basic, and one of the most systematic techniques for developing polynomial time algorithms with overwhelming applications. However, it often suffers from having high running time and space complexity due to (a) maintaining a table of solutions for a large number of sub-instances, and (b) combining/comparing these solutions to successively solve larger sub-instances. In this paper, we consider a canonical cubic time and quadratic space dynamic programming, and show how improvements in both its time and space uses are possible. As a result, we obtain fast small-space approximation algorithms for the fundamental problems of context free grammar recognition} (the basic computer science problem of parsing), the language edit distance} (a significant generalization of string edit distance and parsing), and RNA folding} (a classical problem in bioinformatics). For these problems, ours are the first algorithms that break the cubic-time barrier of any combinatorial algorithm, and quadratic-space barrier of any algorithm significantly improving upon their long-standing space and time complexities. Our technique applies to many other problems as well including string edit distance computation, and finding longest increasing subsequence.Our improvements come from directly grinding the dynamic programming and looking through the lens of language edit distance which generalizes both context free grammar recognition, and RNA folding. From known conditional lower bound results, neither of these problems can have an exact combinatorial algorithm (one that does not use fast matrix multiplication) running in truly subcubic time. Moreover, for language edit distance such an algorithm cannot exist even when nontrivial multiplicative approximation is allowed. We overcome this hurdle by designing an additive-approximation algorithm that for any parameter k ≈ 0, uses O(nklog{n}) space and O(n^2klog{n}) time and provides an additive O(frac{n}{k}log{n})-approximation. In particular, in tilde{O}(n)footnotemark[1] space and tilde{O}(n^2) time it can solve deterministically whether a string belongs to a context free grammar, or ≥ilon-far from it for any constant ≥ilon ≈ 0. We also improve the above results to obtain an algorithm that outputs an ≥ilon⋅ n-additive approximation to the above problems with space complexity O(n^{2/3} log{n}). The space complexity remains sublinear in n, as long as ≥ilon = o(n^{-frac{1}{4}}). Moreover, we provide the first MapReduce and streaming algorithms for them with multiple passes and sublinear space complexity.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124340339","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 show that every H-minor-free graph has a light (1+≥ilon)-spanner, resolving an open problem of Grigni and Sissokho and proving a conjecture of Grigni and Hung cite{GH12}. Our lightness bound is [Oleft(frac{sigma_H}{≥ilon^3}log frac{1}{≥ilon}right)] where sigma_H = |V(H)|√{log |V(H)|} is the sparsity coefficient of H-minor-free graphs. That is, it has a practical dependency on the size of the minor H. Our result also implies that the polynomial time approximation scheme (PTAS) for the Travelling Salesperson Problem (TSP) in H-minor-free graphs by Demaine, Hajiaghayi and Kawarabayashi is an efficient PTAS whose running time is 2^{O_Hleft(frac{1}{≥ilon^4}log frac{1}{≥ilon}right)}n^{O(1)} where O_H ignores dependencies on the size of H. Our techniques significantly deviate from existing lines of research on spanners for H-minor-free graphs, but build upon the work of Chechik and Wulff-Nilsen for spanners of general graphs[6].
{"title":"Minor-Free Graphs Have Light Spanners","authors":"G. Borradaile, Hung Le, Christian Wulff-Nilsen","doi":"10.1109/FOCS.2017.76","DOIUrl":"https://doi.org/10.1109/FOCS.2017.76","url":null,"abstract":"We show that every H-minor-free graph has a light (1+≥ilon)-spanner, resolving an open problem of Grigni and Sissokho and proving a conjecture of Grigni and Hung cite{GH12}. Our lightness bound is [Oleft(frac{sigma_H}{≥ilon^3}log frac{1}{≥ilon}right)] where sigma_H = |V(H)|√{log |V(H)|} is the sparsity coefficient of H-minor-free graphs. That is, it has a practical dependency on the size of the minor H. Our result also implies that the polynomial time approximation scheme (PTAS) for the Travelling Salesperson Problem (TSP) in H-minor-free graphs by Demaine, Hajiaghayi and Kawarabayashi is an efficient PTAS whose running time is 2^{O_Hleft(frac{1}{≥ilon^4}log frac{1}{≥ilon}right)}n^{O(1)} where O_H ignores dependencies on the size of H. Our techniques significantly deviate from existing lines of research on spanners for H-minor-free graphs, but build upon the work of Chechik and Wulff-Nilsen for spanners of general graphs[6].","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114510361","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}
Given a set of points P⊄ R^d and a kernel k, the Kernel Density Estimate at a point x∊R^d is defined as mathrm{KDE}_{P}(x)=frac{1}{|P|}sum_{yin P} k(x,y). We study the problem of designing a data structure that given a data set P and a kernel function, returns approximations to the kernel density} of a query point in sublinear time}. We introduce a class of unbiased estimators for kernel density implemented through locality-sensitive hashing, and give general theorems bounding the variance of such estimators. These estimators give rise to efficient data structures for estimating the kernel density in high dimensions for a variety of commonly used kernels. Our work is the first to provide data-structures with theoretical guarantees that improve upon simple random sampling in high dimensions.
给定一组点P＆＃x2284;R^d和一个核k，在点x＆＃x220A;R^d的核密度估计定义为mathrm{KDE} ＿P{(x)= }frac{1}{|P|}sum ＿y{in P }k(x,y)。我们研究了设计一个数据结构的问题，给定一个数据集P和一个核函数，返回查询点在亚线性时间内的核密度的近似值。我们引入了一类通过位置敏感哈希实现的核密度无偏估计，并给出了该类估计方差的一般定理。这些估计器为估计各种常用核的高维核密度提供了有效的数据结构。我们的工作是第一个提供具有理论保证的数据结构，改进了高维的简单随机抽样。
{"title":"Hashing-Based-Estimators for Kernel Density in High Dimensions","authors":"M. Charikar, Paris Siminelakis","doi":"10.1109/FOCS.2017.99","DOIUrl":"https://doi.org/10.1109/FOCS.2017.99","url":null,"abstract":"Given a set of points P⊄ R^d and a kernel k, the Kernel Density Estimate at a point x∊R^d is defined as mathrm{KDE}_{P}(x)=frac{1}{|P|}sum_{yin P} k(x,y). We study the problem of designing a data structure that given a data set P and a kernel function, returns approximations to the kernel density} of a query point in sublinear time}. We introduce a class of unbiased estimators for kernel density implemented through locality-sensitive hashing, and give general theorems bounding the variance of such estimators. These estimators give rise to efficient data structures for estimating the kernel density in high dimensions for a variety of commonly used kernels. Our work is the first to provide data-structures with theoretical guarantees that improve upon simple random sampling in high dimensions.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127364659","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 a general time-space lower bound that applies for a large class of learning problems and shows that for every problem in that class, any learning algorithm requires either a memory of quadratic size or an exponential number of samples. As a special case, this gives a new proof for the time-space lower bound for parity learning [R16]. Our result is stated in terms of the norm of the matrix that corresponds to the learning problem. Let X, A be two finite sets. Let M: A × X rightarrow {-1,1} be a matrix. The matrix M corresponds to the following learning problem: An unknown element x ∊ X was chosen uniformly at random. A learner tries to learn x from a stream of samples, (a_1, b_1), (a_2, b_2)..., where for every i, a_i ∊ A is chosen uniformly at random and b_i = M(a_i,x). Let sigma be the largest singular value of M and note that always sigma ≤ |A|^{1/2} ⋅ |X|^{1/2}. We show that if sigma ≤ |A|^{1/2} ⋅ |X|^{1/2 - ≥ilon, then any learning algorithm for the corresponding learning problem requires either a memory of size quadratic in ≥ilon n or number of samples exponential in ≥ilon n, where n = log_2 |X|.As a special case, this gives a new proof for the memorysamples lower bound for parity learning [14].
{"title":"A Time-Space Lower Bound for a Large Class of Learning Problems","authors":"R. Raz","doi":"10.1109/FOCS.2017.73","DOIUrl":"https://doi.org/10.1109/FOCS.2017.73","url":null,"abstract":"We prove a general time-space lower bound that applies for a large class of learning problems and shows that for every problem in that class, any learning algorithm requires either a memory of quadratic size or an exponential number of samples. As a special case, this gives a new proof for the time-space lower bound for parity learning [R16]. Our result is stated in terms of the norm of the matrix that corresponds to the learning problem. Let X, A be two finite sets. Let M: A × X rightarrow {-1,1} be a matrix. The matrix M corresponds to the following learning problem: An unknown element x ∊ X was chosen uniformly at random. A learner tries to learn x from a stream of samples, (a_1, b_1), (a_2, b_2)..., where for every i, a_i ∊ A is chosen uniformly at random and b_i = M(a_i,x). Let sigma be the largest singular value of M and note that always sigma ≤ |A|^{1/2} ⋅ |X|^{1/2}. We show that if sigma ≤ |A|^{1/2} ⋅ |X|^{1/2 - ≥ilon, then any learning algorithm for the corresponding learning problem requires either a memory of size quadratic in ≥ilon n or number of samples exponential in ≥ilon n, where n = log_2 |X|.As a special case, this gives a new proof for the memorysamples lower bound for parity learning [14].","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"211 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122456024","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}
Waldo Gálvez, F. Grandoni, Sandy Heydrich, Salvatore Ingala, A. Khan, Andreas Wiese
We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is 2+ε [Jansen and Zhang, SODA 2004]. In this paper we break the 2 approximation barrier, achieving a polynomialtime 17/9 + ε
我们研究了二维几何背包问题(2DK)，在这个问题中，我们得到了一组n个轴对齐的矩形物品，每个物品都有一个相关的利润，以及一个轴对齐的方形背包。目标是找到背包内物品(不旋转物品)的最大利润子集的(非重叠)包装。这个问题最著名的多项式时间近似因子(即使只是在基数情况下)是2+ε[Jansen and Zhang, SODA, 2004]。在本文中，我们打破了2近似障碍，实现了多项式时间17/9 + ε
{"title":"Approximating Geometric Knapsack via L-Packings","authors":"Waldo Gálvez, F. Grandoni, Sandy Heydrich, Salvatore Ingala, A. Khan, Andreas Wiese","doi":"10.1109/FOCS.2017.32","DOIUrl":"https://doi.org/10.1109/FOCS.2017.32","url":null,"abstract":"We study the two-dimensional geometric knapsack problem (2DK) in which we are given a set of n axis-aligned rectangular items, each one with an associated profit, and an axis-aligned square knapsack. The goal is to find a (non-overlapping) packing of a maximum profit subset of items inside the knapsack (without rotating items). The best-known polynomial-time approximation factor for this problem (even just in the cardinality case) is 2+ε [Jansen and Zhang, SODA 2004]. In this paper we break the 2 approximation barrier, achieving a polynomialtime 17/9 + ε","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"62 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121179461","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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