Alexandr Andoni, A. Naor, Aleksandar Nikolov, Ilya P. Razenshteyn, Erik Waingarten
We establish a generic reduction from _nonlinear spectral gaps_ of metric spaces to data-dependent Locality-Sensitive Hashing, yielding a new approach to the high-dimensional Approximate Near Neighbor Search problem (ANN) under various distance functions. Using this reduction, we obtain the following results: * For _general_ d-dimensional normed spaces and n-point datasets, we obtain a _cell-probe_ ANN data structure with approximation O(logd/ε2), space dO(1) n1+ε, and dO(1)nε cell probes per query, for any ε>0. No non-trivial approximation was known before in this generality other than the O(√d) bound which follows from embedding a general norm into ℓ2. * For ℓp and Schatten-p norms, we improve the data structure further, to obtain approximation O(p) and sublinear query _time_. For ℓp, this improves upon the previous best approximation 2O(p) (which required polynomial as opposed to near-linear in n space). For the Schatten-p norm, no non-trivial ANN data structure was known before this work. Previous approaches to the ANN problem either exploit the low dimensionality of a metric, requiring space exponential in the dimension, or circumvent the curse of dimensionality by embedding a metric into a ”tractable” space, such as ℓ1. Our new generic reduction proceeds differently from both of these approaches using a novel partitioning method.
{"title":"Data-dependent hashing via nonlinear spectral gaps","authors":"Alexandr Andoni, A. Naor, Aleksandar Nikolov, Ilya P. Razenshteyn, Erik Waingarten","doi":"10.1145/3188745.3188846","DOIUrl":"https://doi.org/10.1145/3188745.3188846","url":null,"abstract":"We establish a generic reduction from _nonlinear spectral gaps_ of metric spaces to data-dependent Locality-Sensitive Hashing, yielding a new approach to the high-dimensional Approximate Near Neighbor Search problem (ANN) under various distance functions. Using this reduction, we obtain the following results: * For _general_ d-dimensional normed spaces and n-point datasets, we obtain a _cell-probe_ ANN data structure with approximation O(logd/ε2), space dO(1) n1+ε, and dO(1)nε cell probes per query, for any ε>0. No non-trivial approximation was known before in this generality other than the O(√d) bound which follows from embedding a general norm into ℓ2. * For ℓp and Schatten-p norms, we improve the data structure further, to obtain approximation O(p) and sublinear query _time_. For ℓp, this improves upon the previous best approximation 2O(p) (which required polynomial as opposed to near-linear in n space). For the Schatten-p norm, no non-trivial ANN data structure was known before this work. Previous approaches to the ANN problem either exploit the low dimensionality of a metric, requiring space exponential in the dimension, or circumvent the curse of dimensionality by embedding a metric into a ”tractable” space, such as ℓ1. Our new generic reduction proceeds differently from both of these approaches using a novel partitioning method.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74885773","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 describe a new way of compressing two-party communication protocols to get protocols with potentially smaller communication. We show that every communication protocol that communicates C bits and reveals I bits of information about the participants’ private inputs to an observer that watches the communication, can be simulated by a new protocol that communicates at most poly(I) · loglog(C) bits. Our result is tight up to polynomial factors, as it matches the recent work separating communication complexity from external information cost.
{"title":"Interactive compression to external information","authors":"M. Braverman, Gillat Kol","doi":"10.1145/3188745.3188956","DOIUrl":"https://doi.org/10.1145/3188745.3188956","url":null,"abstract":"We describe a new way of compressing two-party communication protocols to get protocols with potentially smaller communication. We show that every communication protocol that communicates C bits and reveals I bits of information about the participants’ private inputs to an observer that watches the communication, can be simulated by a new protocol that communicates at most poly(I) · loglog(C) bits. Our result is tight up to polynomial factors, as it matches the recent work separating communication complexity from external information cost.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75483300","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}
Nisan (Combinatorica’92) constructed a pseudorandom generator for length n, width n read-once branching programs (ROBPs) with error ε and seed length O(log2n + logn · log(1/ε)). A major goal in complexity theory is to reduce the seed length, hopefully, to the optimal O(logn+log(1/ε)), or to construct improved hitting sets, as these would yield stronger derandomization of BPL and RL, respectively. In contrast to a successful line of work in restricted settings, no progress has been made for general, unrestricted, ROBPs. Indeed, Nisan’s construction is the best pseudorandom generator and, prior to this work, also the best hitting set for unrestricted ROBPs. In this work, we make the first improvement for the general case by constructing a hitting set with seed length O(log2n+log(1/ε)). That is, we decouple ε and n, and obtain near-optimal dependence on the former. The regime of parameters in which our construction strictly improves upon prior works, namely, log(1/ε) ≫ logn, is well-motivated by the work of Saks and Zhou (J.CSS’99) who use pseudorandom generators with error ε = 2−(logn)2 in their proof for BPL ⊆ L3/2. We further suggest a research program towards proving that BPL ⊆ L4/3 in which our result achieves one step. As our main technical tool, we introduce and construct a new type of primitive we call pseudorandom pseudo-distributions. Informally, this is a generalization of pseudorandom generators in which one may assign negative and unbounded weights to paths as opposed to working with probability distributions. We show that such a primitive yields hitting sets and, for derandomization purposes, can be used to derandomize two-sided error algorithms.
{"title":"Hitting sets with near-optimal error for read-once branching programs","authors":"M. Braverman, Gil Cohen, Sumegha Garg","doi":"10.1145/3188745.3188780","DOIUrl":"https://doi.org/10.1145/3188745.3188780","url":null,"abstract":"Nisan (Combinatorica’92) constructed a pseudorandom generator for length n, width n read-once branching programs (ROBPs) with error ε and seed length O(log2n + logn · log(1/ε)). A major goal in complexity theory is to reduce the seed length, hopefully, to the optimal O(logn+log(1/ε)), or to construct improved hitting sets, as these would yield stronger derandomization of BPL and RL, respectively. In contrast to a successful line of work in restricted settings, no progress has been made for general, unrestricted, ROBPs. Indeed, Nisan’s construction is the best pseudorandom generator and, prior to this work, also the best hitting set for unrestricted ROBPs. In this work, we make the first improvement for the general case by constructing a hitting set with seed length O(log2n+log(1/ε)). That is, we decouple ε and n, and obtain near-optimal dependence on the former. The regime of parameters in which our construction strictly improves upon prior works, namely, log(1/ε) ≫ logn, is well-motivated by the work of Saks and Zhou (J.CSS’99) who use pseudorandom generators with error ε = 2−(logn)2 in their proof for BPL ⊆ L3/2. We further suggest a research program towards proving that BPL ⊆ L4/3 in which our result achieves one step. As our main technical tool, we introduce and construct a new type of primitive we call pseudorandom pseudo-distributions. Informally, this is a generalization of pseudorandom generators in which one may assign negative and unbounded weights to paths as opposed to working with probability distributions. We show that such a primitive yields hitting sets and, for derandomization purposes, can be used to derandomize two-sided error algorithms.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85211706","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 complexity of Philip Wolfe’s method for the minimum Euclidean-norm point problem over a convex polytope has remained unknown since he proposed the method in 1974. We present the first example that Wolfe’s method takes exponential time. Additionally, we improve previous results to show that linear programming reduces in strongly-polynomial time to the minimum norm point problem over a simplex
Philip Wolfe的凸多面体上最小欧几里得范数点问题的方法自1974年提出以来，其复杂性一直是未知的。我们给出了Wolfe方法需要指数时间的第一个例子。此外，我们改进了先前的结果，表明线性规划在强多项式时间内减少了单纯形上的最小范数点问题
{"title":"The minimum Euclidean-norm point in a convex polytope: Wolfe's combinatorial algorithm is exponential","authors":"J. D. Loera, Jamie Haddock, Luis Rademacher","doi":"10.1145/3188745.3188820","DOIUrl":"https://doi.org/10.1145/3188745.3188820","url":null,"abstract":"The complexity of Philip Wolfe’s method for the minimum Euclidean-norm point problem over a convex polytope has remained unknown since he proposed the method in 1974. We present the first example that Wolfe’s method takes exponential time. Additionally, we improve previous results to show that linear programming reduces in strongly-polynomial time to the minimum norm point problem over a simplex","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80323550","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}
This paper proves that the welfare of the first price auction in Bayes-Nash equilibrium is at least a .743-fraction of the welfare of the optimal mechanism assuming agents’ values are independently distributed. The previous best bound was 1−1/e≈.63, derived using smoothness, the standard technique for reasoning about welfare of games in equilibrium. In the worst known example, the first price auction achieves a ≈.869-fraction of the optimal welfare, far better than the theoretical guarantee. Despite this large gap, it was unclear whether the 1−1/e bound was tight. We prove that it is not. Our analysis eschews smoothness, and instead uses the independence assumption on agents’ value distributions to give a more careful accounting of the welfare contribution of agents who win despite not having the highest value.
{"title":"A tighter welfare guarantee for first-price auctions","authors":"D. Hoy, Sam Taggart, Zihe Wang","doi":"10.1145/3188745.3188944","DOIUrl":"https://doi.org/10.1145/3188745.3188944","url":null,"abstract":"This paper proves that the welfare of the first price auction in Bayes-Nash equilibrium is at least a .743-fraction of the welfare of the optimal mechanism assuming agents’ values are independently distributed. The previous best bound was 1−1/e≈.63, derived using smoothness, the standard technique for reasoning about welfare of games in equilibrium. In the worst known example, the first price auction achieves a ≈.869-fraction of the optimal welfare, far better than the theoretical guarantee. Despite this large gap, it was unclear whether the 1−1/e bound was tight. We prove that it is not. Our analysis eschews smoothness, and instead uses the independence assumption on agents’ value distributions to give a more careful accounting of the welfare contribution of agents who win despite not having the highest value.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81440185","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 an agent-based distributed algorithm with exponentially distributed waiting times in which agents with binary states interact locally over a geometric graph, and based on this interaction and on the value of a common intolerance threshold τ, decide whether to change their states. This model is equivalent to an Asynchronous Cellular Automaton (ACA) with extended Moore neighborhoods, a zero-temperature Ising model with Glauber dynamics, or a Schelling model of self-organized segregation in an open system, and has applications in the analysis of social and biological networks, and spin glasses systems. We prove a shape theorem for the spread of the “affected” nodes during the process dynamics and show that in the steady state, for τ ∈ (τ*,1−τ*) ∖ {1/2}, where τ* ≈ 0.488, the size of the “mono-chromatic region” at the end of the process is at least exponential in the size of the local neighborhood of interaction with probability approaching one as N grows. Combined with previous results on the expected size of the monochromatic region that provide a matching upper bound, this implies that in the steady state the size of the monochromatic region of any agent is exponential with high probability for the mentioned interval of τ. The shape theorem is based on a novel concentration inequality for the spreading time, and provides a precise geometrical description of the process dynamics. The result on the size of the monochromatic region considerably extends our understanding of the steady state. Showing convergence with high probability, it rules out the possibility that only a small fraction of the nodes are eventually contained in large monochromatic regions, which was left open by previous works.
{"title":"Shape of diffusion and size of monochromatic region of a two-dimensional spin system","authors":"H. Omidvar, M. Franceschetti","doi":"10.1145/3188745.3188836","DOIUrl":"https://doi.org/10.1145/3188745.3188836","url":null,"abstract":"We consider an agent-based distributed algorithm with exponentially distributed waiting times in which agents with binary states interact locally over a geometric graph, and based on this interaction and on the value of a common intolerance threshold τ, decide whether to change their states. This model is equivalent to an Asynchronous Cellular Automaton (ACA) with extended Moore neighborhoods, a zero-temperature Ising model with Glauber dynamics, or a Schelling model of self-organized segregation in an open system, and has applications in the analysis of social and biological networks, and spin glasses systems. We prove a shape theorem for the spread of the “affected” nodes during the process dynamics and show that in the steady state, for τ ∈ (τ*,1−τ*) ∖ {1/2}, where τ* ≈ 0.488, the size of the “mono-chromatic region” at the end of the process is at least exponential in the size of the local neighborhood of interaction with probability approaching one as N grows. Combined with previous results on the expected size of the monochromatic region that provide a matching upper bound, this implies that in the steady state the size of the monochromatic region of any agent is exponential with high probability for the mentioned interval of τ. The shape theorem is based on a novel concentration inequality for the spreading time, and provides a precise geometrical description of the process dynamics. The result on the size of the monochromatic region considerably extends our understanding of the steady state. Showing convergence with high probability, it rules out the possibility that only a small fraction of the nodes are eventually contained in large monochromatic regions, which was left open by previous works.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87060415","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study the adaptive complexity of submodular optimization. Informally, the adaptive complexity of a problem is the minimal number of sequential rounds required to achieve a constant factor approximation when polynomially-many queries can be executed in parallel at each round. Adaptivity is a fundamental concept that is heavily studied in computer science, largely due to the need for parallelizing computation. Somewhat surprisingly, very little is known about adaptivity in submodular optimization. For the canonical problem of maximizing a monotone submodular function under a cardinality constraint, to the best of our knowledge, all that is known to date is that the adaptive complexity is between 1 and Ω(n). Our main result in this paper is a tight characterization showing that the adaptive complexity of maximizing a monotone submodular function under a cardinality constraint is Θ(log n): - We describe an algorithm which requires O(log n) sequential rounds and achieves an approximation that is arbitrarily close to 1/3; - We show that no algorithm can achieve an approximation better than O(1 / log n) with fewer than O(log n / log log n) rounds. Thus, when allowing for parallelization, our algorithm achieves a constant factor approximation exponentially faster than any known existing algorithm for submodular maximization. Importantly, the approximation algorithm is achieved via adaptive sampling and complements a recent line of work on optimization of functions learned from data. In many cases we do not know the functions we optimize and learn them from labeled samples. Recent results show that no algorithm can obtain a constant factor approximation guarantee using polynomially-many labeled samples as in the PAC and PMAC models, drawn from any distribution. Since learning with non-adaptive samples over any distribution results in a sharp impossibility, we consider learning with adaptive samples where the learner obtains poly(n) samples drawn from a distribution of her choice in every round. Our result implies that in the realizable case, where there is a true underlying function generating the data, Θ(log n) batches of adaptive samples are necessary and sufficient to approximately “learn to optimize” a monotone submodular function under a cardinality constraint.
{"title":"The adaptive complexity of maximizing a submodular function","authors":"Eric Balkanski, Yaron Singer","doi":"10.1145/3188745.3188752","DOIUrl":"https://doi.org/10.1145/3188745.3188752","url":null,"abstract":"In this paper we study the adaptive complexity of submodular optimization. Informally, the adaptive complexity of a problem is the minimal number of sequential rounds required to achieve a constant factor approximation when polynomially-many queries can be executed in parallel at each round. Adaptivity is a fundamental concept that is heavily studied in computer science, largely due to the need for parallelizing computation. Somewhat surprisingly, very little is known about adaptivity in submodular optimization. For the canonical problem of maximizing a monotone submodular function under a cardinality constraint, to the best of our knowledge, all that is known to date is that the adaptive complexity is between 1 and Ω(n). Our main result in this paper is a tight characterization showing that the adaptive complexity of maximizing a monotone submodular function under a cardinality constraint is Θ(log n): - We describe an algorithm which requires O(log n) sequential rounds and achieves an approximation that is arbitrarily close to 1/3; - We show that no algorithm can achieve an approximation better than O(1 / log n) with fewer than O(log n / log log n) rounds. Thus, when allowing for parallelization, our algorithm achieves a constant factor approximation exponentially faster than any known existing algorithm for submodular maximization. Importantly, the approximation algorithm is achieved via adaptive sampling and complements a recent line of work on optimization of functions learned from data. In many cases we do not know the functions we optimize and learn them from labeled samples. Recent results show that no algorithm can obtain a constant factor approximation guarantee using polynomially-many labeled samples as in the PAC and PMAC models, drawn from any distribution. Since learning with non-adaptive samples over any distribution results in a sharp impossibility, we consider learning with adaptive samples where the learner obtains poly(n) samples drawn from a distribution of her choice in every round. Our result implies that in the realizable case, where there is a true underlying function generating the data, Θ(log n) batches of adaptive samples are necessary and sufficient to approximately “learn to optimize” a monotone submodular function under a cardinality constraint.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83388415","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}
Zhengyang Liu, Xi Chen, R. Servedio, Ying Sheng, Jinyu Xie
We study the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0,1}n. Our first main result is that distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that uses Õ(k2)/є queries (independent of n). Complementing this, our second main result is a lower bound showing that any non-adaptive distribution-free k-junta testing algorithm must make Ω(2k/3) queries even to test to accuracy є=1/3. These bounds establish that while the optimal query complexity of non-adaptive k-junta testing is 2Θ(k), for adaptive testing it is poly(k), and thus show that adaptivity provides an exponential improvement in the distribution-free query complexity of testing juntas.
{"title":"Distribution-free junta testing","authors":"Zhengyang Liu, Xi Chen, R. Servedio, Ying Sheng, Jinyu Xie","doi":"10.1145/3188745.3188842","DOIUrl":"https://doi.org/10.1145/3188745.3188842","url":null,"abstract":"We study the problem of testing whether an unknown n-variable Boolean function is a k-junta in the distribution-free property testing model, where the distance between functions is measured with respect to an arbitrary and unknown probability distribution over {0,1}n. Our first main result is that distribution-free k-junta testing can be performed, with one-sided error, by an adaptive algorithm that uses Õ(k2)/є queries (independent of n). Complementing this, our second main result is a lower bound showing that any non-adaptive distribution-free k-junta testing algorithm must make Ω(2k/3) queries even to test to accuracy є=1/3. These bounds establish that while the optimal query complexity of non-adaptive k-junta testing is 2Θ(k), for adaptive testing it is poly(k), and thus show that adaptivity provides an exponential improvement in the distribution-free query complexity of testing juntas.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82025559","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}
Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, S. Safra
We present a polynomial time reduction from gap-3LIN to label cover with 2-to-1 constraints. In the “yes” case the fraction of satisfied constraints is at least 1 −ε, and in the “no” case we show that this fraction is at most ε, assuming a certain (new) combinatorial hypothesis on the Grassmann graph. In other words, we describe a combinatorial hypothesis that implies the 2-to-1 conjecture with imperfect completeness. The companion submitted paper [Dinur, Khot, Kindler, Minzer and Safra, STOC 2018] makes some progress towards proving this hypothesis. Our work builds on earlier work by a subset of the authors [Khot, Minzer and Safra, STOC 2017] where a slightly different hypothesis was used to obtain hardness of approximating vertex cover to within factor of √2−ε. The most important implication of this work is (assuming the hypothesis) an NP-hardness gap of 1/2−ε vs. ε for unique games. In addition, we derive optimal NP-hardness for approximating the max-cut-gain problem, NP-hardness of coloring an almost 4-colorable graph with any constant number of colors, and the same √2−ε NP-hardness for approximate vertex cover that was already obtained based on a slightly different hypothesis. Recent progress towards proving our hypothesis [Barak, Kothari and Steurer, ECCC TR18-077], [Dinur, Khot, Kindler, Minzer and Safra, STOC 2018] directly implies some new unconditional NP-hardness results. These include new points of NP-hardness for unique games and for 2-to-1 and 2-to-2 games. More recently, the full version of our hypothesis was proven [Khot, Minzer and Safra, ECCC TR18-006].
{"title":"Towards a proof of the 2-to-1 games conjecture?","authors":"Irit Dinur, Subhash Khot, Guy Kindler, Dor Minzer, S. Safra","doi":"10.1145/3188745.3188804","DOIUrl":"https://doi.org/10.1145/3188745.3188804","url":null,"abstract":"We present a polynomial time reduction from gap-3LIN to label cover with 2-to-1 constraints. In the “yes” case the fraction of satisfied constraints is at least 1 −ε, and in the “no” case we show that this fraction is at most ε, assuming a certain (new) combinatorial hypothesis on the Grassmann graph. In other words, we describe a combinatorial hypothesis that implies the 2-to-1 conjecture with imperfect completeness. The companion submitted paper [Dinur, Khot, Kindler, Minzer and Safra, STOC 2018] makes some progress towards proving this hypothesis. Our work builds on earlier work by a subset of the authors [Khot, Minzer and Safra, STOC 2017] where a slightly different hypothesis was used to obtain hardness of approximating vertex cover to within factor of √2−ε. The most important implication of this work is (assuming the hypothesis) an NP-hardness gap of 1/2−ε vs. ε for unique games. In addition, we derive optimal NP-hardness for approximating the max-cut-gain problem, NP-hardness of coloring an almost 4-colorable graph with any constant number of colors, and the same √2−ε NP-hardness for approximate vertex cover that was already obtained based on a slightly different hypothesis. Recent progress towards proving our hypothesis [Barak, Kothari and Steurer, ECCC TR18-077], [Dinur, Khot, Kindler, Minzer and Safra, STOC 2018] directly implies some new unconditional NP-hardness results. These include new points of NP-hardness for unique games and for 2-to-1 and 2-to-2 games. More recently, the full version of our hypothesis was proven [Khot, Minzer and Safra, ECCC TR18-006].","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79657507","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}
M. Bateni, M. Hajiaghayi, Saeed Seddighin, Cliff Stein
The knapsack problem is a fundamental problem in combinatorial optimization. It has been studied extensively from theoretical as well as practical perspectives as it is one of the most well-known NP-hard problems. The goal is to pack a knapsack of size t with the maximum value from a collection of n items with given sizes and values. Recent evidence suggests that a classic O(nt) dynamic-programming solution for the knapsack problem might be the fastest in the worst case. In fact, solving the knapsack problem was shown to be computationally equivalent to the (min, +) convolution problem, which is thought to be facing a quadratic-time barrier. This hardness is in contrast to the more famous (+, ·) convolution (generally known as polynomial multiplication), that has an O(nlogn)-time solution via Fast Fourier Transform. Our main results are algorithms with near-linear running times (in terms of the size of the knapsack and the number of items) for the knapsack problem, if either the values or sizes of items are small integers. More specifically, if item sizes are integers bounded by , the running time of our algorithm is Õ((n+t)). If the item values are integers bounded by , our algorithm runs in time Õ(n+t). Best previously known running times were O(nt), O(n2) and O(n) (Pisinger, J. of Alg., 1999). At the core of our algorithms lies the prediction technique: Roughly speaking, this new technique enables us to compute the convolution of two vectors in time (n) when an approximation of the solution within an additive error of is available. Our results also improve the best known strongly polynomial time solutions for knapsack. In the limited size setting, when the items have multiplicities, the fastest strongly polynomial time algorithms for knapsack run in time O(n2 2) and O(n3 2) for the cases of infinite and given multiplicities, respectively. Our results improve both running times by a factor of (n max{1, n/}).
背包问题是组合优化中的一个基本问题。由于它是最著名的np困难问题之一，从理论和实践的角度对它进行了广泛的研究。目标是用给定大小和值的n个物品集合中的最大值打包一个大小为t的背包。最近的证据表明，在最坏的情况下，背包问题的经典O(nt)动态规划解决方案可能是最快的。事实上，解决背包问题被证明在计算上等同于(min， +)卷积问题，后者被认为面临二次时间障碍。这种硬度与更著名的(+，·)卷积(通常称为多项式乘法)形成对比，后者通过快速傅里叶变换具有O(nlogn)时间解决方案。如果物品的值或大小是小整数，我们的主要结果是具有近似线性运行时间(就背包的大小和物品的数量而言)的算法。更具体地说，如果项目大小是整数，则算法的运行时间为Õ((n+t))。如果项值是有边界的整数，我们的算法运行时间为Õ(n+t)。以前已知的最佳运行时间是O(nt)、O(n2)和O(n) (Pisinger, J. of Alg)。, 1999)。我们算法的核心是预测技术:粗略地说，这种新技术使我们能够计算两个向量在时间(n)上的卷积，当在可加性误差范围内的近似解可用时。我们的结果也改进了最著名的强多项式时间解。在有限大小的设置中，当项目具有多重性时，对于无限多重性和给定多重性的情况，最快的强多项式时间背包算法分别在O(n2 2)和O(n32)时间内运行。我们的结果将运行时间提高了(n max{1, n/})倍。
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