The deletion channel takes as input a bit string x ∊ {0,1}^n, and deletes each bit independently with probability q, yielding a shorter string. The trace reconstruction problem is to recover an unknown string x ∊ from many independent outputs (called traces) of the deletion channel applied to x.We show that if x is drawn uniformly at random and q
{"title":"Average-Case Reconstruction for the Deletion Channel: Subpolynomially Many Traces Suffice","authors":"Y. Peres, Alex Zhai","doi":"10.1109/FOCS.2017.29","DOIUrl":"https://doi.org/10.1109/FOCS.2017.29","url":null,"abstract":"The deletion channel takes as input a bit string x ∊ {0,1}^n, and deletes each bit independently with probability q, yielding a shorter string. The trace reconstruction problem is to recover an unknown string x ∊ from many independent outputs (called traces) of the deletion channel applied to x.We show that if x is drawn uniformly at random and q","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125978151","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study approximation algorithms for scheduling problems with the objective of minimizing total weighted completion time, under identical and related machine models with job precedence constraints. We give algorithms that improve upon many previous 15 to 20-year-old state-of-art results. A major theme in these results is the use of time-indexed linear programming relaxations. These are natural relaxations for their respective problems, but surprisingly are not studied in the literature.We also consider the scheduling problem of minimizing total weighted completion time on unrelated machines. The recent breakthrough result of [Bansal-Srinivasan-Svensson, STOC 2016] gave a (1.5-c)-approximation for the problem, based on some lift-and-project SDP relaxation. Our main result is that a (1.5 - c)-approximation can also be achieved using a natural and considerably simpler time-indexed LP relaxation for the problem. We hope this relaxation can provide new insights into the problem.
{"title":"Scheduling to Minimize Total Weighted Completion Time via Time-Indexed Linear Programming Relaxations","authors":"Shi Li","doi":"10.1109/FOCS.2017.34","DOIUrl":"https://doi.org/10.1109/FOCS.2017.34","url":null,"abstract":"We study approximation algorithms for scheduling problems with the objective of minimizing total weighted completion time, under identical and related machine models with job precedence constraints. We give algorithms that improve upon many previous 15 to 20-year-old state-of-art results. A major theme in these results is the use of time-indexed linear programming relaxations. These are natural relaxations for their respective problems, but surprisingly are not studied in the literature.We also consider the scheduling problem of minimizing total weighted completion time on unrelated machines. The recent breakthrough result of [Bansal-Srinivasan-Svensson, STOC 2016] gave a (1.5-c)-approximation for the problem, based on some lift-and-project SDP relaxation. Our main result is that a (1.5 - c)-approximation can also be achieved using a natural and considerably simpler time-indexed LP relaxation for the problem. We hope this relaxation can provide new insights into the problem.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116161074","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}
Felix Joos, Jaehoon Kim, Daniela Kühn, Deryk Osthus
We provide a combinatorial characterization of all testable properties of k-graphs (i.e. k-uniform hypergraphs). Here, a k-graph property P is testable if there is a randomized algorithm which makes a bounded number of edge queries and distinguishes with probability 2/3 between k-graphs that satisfy P and those that are far from satisfying P. For the 2-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. Our results for the k-graph setting are in contrast to those of Austin and Tao, who showed that for the somewhat stronger concept of local repairability, the testability results for graphs do not extend to the 3-graph setting.
{"title":"A Characterization of Testable Hypergraph Properties","authors":"Felix Joos, Jaehoon Kim, Daniela Kühn, Deryk Osthus","doi":"10.1109/FOCS.2017.84","DOIUrl":"https://doi.org/10.1109/FOCS.2017.84","url":null,"abstract":"We provide a combinatorial characterization of all testable properties of k-graphs (i.e. k-uniform hypergraphs). Here, a k-graph property P is testable if there is a randomized algorithm which makes a bounded number of edge queries and distinguishes with probability 2/3 between k-graphs that satisfy P and those that are far from satisfying P. For the 2-graph case, such a combinatorial characterization was obtained by Alon, Fischer, Newman and Shapira. Our results for the k-graph setting are in contrast to those of Austin and Tao, who showed that for the somewhat stronger concept of local repairability, the testability results for graphs do not extend to the 3-graph setting.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129230993","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}
Several fundamental problems that arise in optimization and computer science can be cast as follows: Given vectors v_1,...,v_m in R^d and a constraint family B of subsets of [m], find a set S in B that maximizes the squared volume of the simplex spanned by the vectors in S. A motivating example is the ubiquitous data-summarization problem in machine learning and information retrieval where one is given a collection of feature vectors that represent data such as documents or images. The volume of a collection of vectors is used as a measure of their diversity, and partition or matroid constraints over [m] are imposed in order to ensure resource or fairness constraints. Even with a simple cardinality constraint, the problem becomes NP-hard and has received much attention starting with a result by Khachiyan who gave an r^{O(r)} approximation algorithm for this problem. Recently, Nikolov and Singh presented a convex program and showed how it can be used to estimate the value of the most diverse set when there are multiple cardinality constraints (i.e., when B corresponds to a partition matroid). Their proof of the integrality gap of the convex program relied on an inequality by Gurvits, and was recently extended to regular matroids. The question of whether these estimation algorithms can be converted into the more useful approximation algorithms – that also output a set – remained open.The main contribution of this paper is to give the first approximation algorithms for both partition and regular matroids. We present novel formulations for the subdeterminant maximization problem for these matroids; this reduces them to the problem of finding a point that maximizes the absolute value of a nonconvex function over a Cartesian product of probability simplices. The technical core of our results is a new anti-concentration inequality for dependent random variables that arise from these functions which allows us to relate the optimal value of these nonconvex functions to their value at a random point. Unlike prior work on the constrained subdeterminant maximization problem, our proofs do not rely on real-stability or convexity and could be of independent interest both in algorithms and complexity where anti-concentration phenomena has recently been deployed.
{"title":"Subdeterminant Maximization via Nonconvex Relaxations and Anti-Concentration","authors":"J. Ebrahimi, D. Straszak, Nisheeth K. Vishnoi","doi":"10.1109/FOCS.2017.98","DOIUrl":"https://doi.org/10.1109/FOCS.2017.98","url":null,"abstract":"Several fundamental problems that arise in optimization and computer science can be cast as follows: Given vectors v_1,...,v_m in R^d and a constraint family B of subsets of [m], find a set S in B that maximizes the squared volume of the simplex spanned by the vectors in S. A motivating example is the ubiquitous data-summarization problem in machine learning and information retrieval where one is given a collection of feature vectors that represent data such as documents or images. The volume of a collection of vectors is used as a measure of their diversity, and partition or matroid constraints over [m] are imposed in order to ensure resource or fairness constraints. Even with a simple cardinality constraint, the problem becomes NP-hard and has received much attention starting with a result by Khachiyan who gave an r^{O(r)} approximation algorithm for this problem. Recently, Nikolov and Singh presented a convex program and showed how it can be used to estimate the value of the most diverse set when there are multiple cardinality constraints (i.e., when B corresponds to a partition matroid). Their proof of the integrality gap of the convex program relied on an inequality by Gurvits, and was recently extended to regular matroids. The question of whether these estimation algorithms can be converted into the more useful approximation algorithms – that also output a set – remained open.The main contribution of this paper is to give the first approximation algorithms for both partition and regular matroids. We present novel formulations for the subdeterminant maximization problem for these matroids; this reduces them to the problem of finding a point that maximizes the absolute value of a nonconvex function over a Cartesian product of probability simplices. The technical core of our results is a new anti-concentration inequality for dependent random variables that arise from these functions which allows us to relate the optimal value of these nonconvex functions to their value at a random point. Unlike prior work on the constrained subdeterminant maximization problem, our proofs do not rely on real-stability or convexity and could be of independent interest both in algorithms and complexity where anti-concentration phenomena has recently been deployed.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124997144","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 new distributed} model of probabilistically checkable proofs (PCP). A satisfying assignment x ∊ {0,1}^n to a CNF formula phi is shared between two parties, where Alice knows x_1, dots, x_{n/2, Bob knows x_{n/2+1},dots,x_n, and both parties know phi. The goal is to have Alice and Bob jointly write a PCP that x satisfies phi, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic} variant, where the players are helped by Merlin, a third party who knows all of x.Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that %(assuming SETH) there are no truly-subquadratic approximation algorithms for %the following problems: Maximum Inner Product over {0,1}-vectors, LCS Closest Pair over permutations, Approximate Partial Match, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first three problems we obtain nearly-polynomial factors of 2^{(log n)^{1-o(1)}};only (1+o(1))-factor lower bounds (under SETH) were known before.As an additional feature of our reduction, we obtain new SETH lower bounds for the exact} monochromatic Closest Pair problem in the Euclidean, Manhattan, and Hamming metrics.
提出了一种新的概率可检验证明(PCP)分布式模型。一个令人满意的作业x ＆＃x220A;｛0,1｝^n的CNF公式phi在双方之间共享，其中Alice知道x＿1, dots,{x＿n /2, Bob知道x＿n{/2+1, }dots,x＿n，双方都知道phi。目标是让Alice和Bob共同编写x满足phi的PCP，同时交换很少或不交换信息。不幸的是，这个模型不允许非常复杂的查询。相反，我们专注于一个非确定性的}变体，其中玩家得到梅林的帮助，梅林是一个知道所有x的第三方。使用我们的框架，我们首次获得了从强指数时间假设(SETH)到P近似问题的类似pcp的缩减。特别地，在SETH中我们展示了这一点 %(assuming SETH) there are no truly-subquadratic approximation algorithms for %the following problems: Maximum Inner Product over {0,1}-vectors, LCS Closest Pair over permutations, Approximate Partial Match, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first three problems we obtain nearly-polynomial factors of 2^{(log n)^{1-o(1)}};only (1+o(1))-factor lower bounds (under SETH) were known before.As an additional feature of our reduction, we obtain new SETH lower bounds for the exact} monochromatic Closest Pair problem in the Euclidean, Manhattan, and Hamming metrics.
{"title":"Distributed PCP Theorems for Hardness of Approximation in P","authors":"Amir Abboud, A. Rubinstein, Richard Ryan Williams","doi":"10.1109/FOCS.2017.12","DOIUrl":"https://doi.org/10.1109/FOCS.2017.12","url":null,"abstract":"We present a new distributed} model of probabilistically checkable proofs (PCP). A satisfying assignment x ∊ {0,1}^n to a CNF formula phi is shared between two parties, where Alice knows x_1, dots, x_{n/2, Bob knows x_{n/2+1},dots,x_n, and both parties know phi. The goal is to have Alice and Bob jointly write a PCP that x satisfies phi, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic} variant, where the players are helped by Merlin, a third party who knows all of x.Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that %(assuming SETH) there are no truly-subquadratic approximation algorithms for %the following problems: Maximum Inner Product over {0,1}-vectors, LCS Closest Pair over permutations, Approximate Partial Match, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first three problems we obtain nearly-polynomial factors of 2^{(log n)^{1-o(1)}};only (1+o(1))-factor lower bounds (under SETH) were known before.As an additional feature of our reduction, we obtain new SETH lower bounds for the exact} monochromatic Closest Pair problem in the Euclidean, Manhattan, and Hamming metrics.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129982965","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 a simple, multiplicative-weight update algorithm for learning undirected graphical models or Markov random fields (MRFs). The approach is new, and for the well-studied case of Ising models or Boltzmann machines we obtain an algorithm that uses a nearlyoptimal number of samples and has running time O(n^2) (where n is the dimension), subsuming and improving on all prior work. Additionally, we give the first efficient algorithm for learning Ising models over non-binary alphabets.Our main application is an algorithm for learning the structure of t-wise MRFs with nearly-optimal sample complexity (up to polynomial losses in necessary terms that depend on the weights) and running time that is n^t. In addition, given n^t samples, we can also learn the parameters of the model and generate a hypothesis that is close in statistical distance to the true MRF. All prior work runs in time n^d for graphs of bounded degree d and does not generate a hypothesis close in statistical distance even for t = 3. We observe that our runtime has the correct dependence on n and t assuming the hardness of learning sparse parities with noise.Our algorithm– the Sparsitron– is easy to implement (has only one parameter) and holds in the on-line setting. Its analysis applies a regret bound from Freund and Schapires classic Hedge algorithm. It also gives the first solution to the problem of learning sparse Generalized Linear Models (GLMs).
{"title":"Learning Graphical Models Using Multiplicative Weights","authors":"Adam R. Klivans, R. Meka","doi":"10.1109/FOCS.2017.39","DOIUrl":"https://doi.org/10.1109/FOCS.2017.39","url":null,"abstract":"We give a simple, multiplicative-weight update algorithm for learning undirected graphical models or Markov random fields (MRFs). The approach is new, and for the well-studied case of Ising models or Boltzmann machines we obtain an algorithm that uses a nearlyoptimal number of samples and has running time O(n^2) (where n is the dimension), subsuming and improving on all prior work. Additionally, we give the first efficient algorithm for learning Ising models over non-binary alphabets.Our main application is an algorithm for learning the structure of t-wise MRFs with nearly-optimal sample complexity (up to polynomial losses in necessary terms that depend on the weights) and running time that is n^t. In addition, given n^t samples, we can also learn the parameters of the model and generate a hypothesis that is close in statistical distance to the true MRF. All prior work runs in time n^d for graphs of bounded degree d and does not generate a hypothesis close in statistical distance even for t = 3. We observe that our runtime has the correct dependence on n and t assuming the hardness of learning sparse parities with noise.Our algorithm– the Sparsitron– is easy to implement (has only one parameter) and holds in the on-line setting. Its analysis applies a regret bound from Freund and Schapires classic Hedge algorithm. It also gives the first solution to the problem of learning sparse Generalized Linear Models (GLMs).","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124545987","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we give the first construction of high-rate locally list-recoverable codes. List-recovery has been an extremely useful building block in coding theory, and our motivation is to use these codes as such a building block. In particular, our construction gives the first capacity-achieving locally list-decodable codes (over constant-sized alphabet); the first capacity achieving} globally list-decodable codes with nearly linear time list decoding algorithm (once more, over constant-sized alphabet); and a randomized construction of binary codes on the Gilbert-Varshamov bound that can be uniquely decoded in near-linear-time, with higher rate than was previously known.Our techniques are actually quite simple, and are inspired by an approach of Gopalan, Guruswami, and Raghavendra (Siam Journal on Computing, 2011) for list-decoding tensor codes. We show that tensor powers of (globally) list-recoverable codes are approximately locally list-recoverable, and that the approximately modifier may be removed by pre-encoding the message with a suitable locally decodable code. Instantiating this with known constructions of high-rate globally list-recoverable codes and high-rate locally decodable codes finishes the construction.
在这项工作中，我们首次构造了高速率的局部列表可恢复码。列表恢复是编码理论中非常有用的构建块，我们的动机是将这些代码用作这样的构建块。特别是，我们的构造给出了第一个容量实现局部列表可解码的代码(在恒定大小的字母表上);第一个使用近线性时间列表解码算法实现全局列表可解码代码的容量(同样，在恒定大小的字母表上);在Gilbert-Varshamov边界上随机构造二进制代码，可以在近线性时间内唯一解码，并且比以前已知的速率更高。我们的技术实际上非常简单，并且受到Gopalan, Guruswami和Raghavendra (Siam Journal on Computing, 2011)的列表解码张量代码方法的启发。我们证明了(全局)列表可恢复码的张量幂是近似局部列表可恢复的，并且近似修饰符可以通过用合适的局部可解码码对消息进行预编码来去除。用已知的高速率全局列表可恢复代码和高速率局部可解码代码的构造实例化此代码，完成构造。
{"title":"Local List Recovery of High-Rate Tensor Codes & Applications","authors":"B. Hemenway, Noga Ron-Zewi, Mary Wootters","doi":"10.1109/FOCS.2017.27","DOIUrl":"https://doi.org/10.1109/FOCS.2017.27","url":null,"abstract":"In this work, we give the first construction of high-rate locally list-recoverable codes. List-recovery has been an extremely useful building block in coding theory, and our motivation is to use these codes as such a building block. In particular, our construction gives the first capacity-achieving locally list-decodable codes (over constant-sized alphabet); the first capacity achieving} globally list-decodable codes with nearly linear time list decoding algorithm (once more, over constant-sized alphabet); and a randomized construction of binary codes on the Gilbert-Varshamov bound that can be uniquely decoded in near-linear-time, with higher rate than was previously known.Our techniques are actually quite simple, and are inspired by an approach of Gopalan, Guruswami, and Raghavendra (Siam Journal on Computing, 2011) for list-decoding tensor codes. We show that tensor powers of (globally) list-recoverable codes are approximately locally list-recoverable, and that the approximately modifier may be removed by pre-encoding the message with a suitable locally decodable code. Instantiating this with known constructions of high-rate globally list-recoverable codes and high-rate locally decodable codes finishes the construction.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"102 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121476086","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 repair problem for an (n, k) error-correcting code calls for recovery of an unavailable coordinate of the codeword by downloading as little information as possible from a subset of the remaining coordinates. Using the terminology motivated by coding in distributed storage, we attempt to repair a failed node by accessing information stored on d helper nodes, where k ≼ d ≼ n – 1, and using as little repair bandwidth as possible to recover the lost information.By the so-called cut-set bound (Dimakis et al., 2010), the repair bandwidth of an (n,k = n – r) MDS code using d helper nodes is at least dl/(d + 1 – k), where l is the size of the node. A number of constructions of MDS array codes have been shown to meet this bound with equality. In a related but separate line of work, Guruswami and Wootters (2016) studied repair of Reed-Solomon (RS) codes, showing that it is possible to perform repair using a smaller bandwidth than under the trivial approach. At the same time, their work as well as follow-up papers stopped short of constructing RS codes (or any scalar MDS codes) that meet the cut-set bound with equality, which has been an open problem in coding theory.In this work we present a solution to this problem, constructing RS codes of length n over the field of size (ql, l = exp((1 + o(1)n log n) that meet the cut-set bound. We also prove an almost matching lower bound on l, showing that super-exponential scaling is both necessary and sufficient for achieving the cut-set bound using linear repair schemes. More precisely, we prove that for scalar MDS codes (including the RS codes) to meet this bound, the sub-packetization l must satisfy l ≽ exp((1 + o(1))k log k).
(n, k)纠错码的修复问题要求通过从剩余坐标的子集中下载尽可能少的信息来恢复码字的不可用坐标。使用分布式存储编码所激发的术语，我们试图通过访问存储在d个辅助节点上的信息来修复故障节点，其中k ≼d & # x227C;n & # x2013;1、使用尽可能少的修复带宽恢复丢失的信息。根据所谓的切集界(Dimakis et al.， 2010)， an (n,k = n –r)使用d个辅助节点的MDS代码至少为dl/(d + 1 –K)，其中l为节点的大小。许多MDS阵列码的构造已经被证明是相等地满足这个界限。在一个相关但独立的工作中，Guruswami和wooters(2016)研究了Reed-Solomon (RS)代码的修复，表明可以使用比琐碎方法更小的带宽进行修复。与此同时，他们的工作以及后续的论文都没有构造出满足相等切集界的RS码(或任何标量MDS码)，这一直是编码理论中的一个开放性问题。在本文中，我们给出了一个解决这个问题的方法，在大小为(ql, l = exp((1 + o(1)n log n))的域上构造长度为n的满足切集界的RS码。我们还证明了l上的一个几乎匹配的下界，证明了用线性修复方案实现切集界的超指数标度是充分必要的。更准确地说，我们证明了标量MDS码(包括RS码)要满足这个界，子分组l必须满足l ≽Exp ((1 + 0 (1))k log k)
{"title":"Optimal Repair of Reed-Solomon Codes: Achieving the Cut-Set Bound","authors":"Itzhak Tamo, Min Ye, A. Barg","doi":"10.1109/FOCS.2017.28","DOIUrl":"https://doi.org/10.1109/FOCS.2017.28","url":null,"abstract":"The repair problem for an (n, k) error-correcting code calls for recovery of an unavailable coordinate of the codeword by downloading as little information as possible from a subset of the remaining coordinates. Using the terminology motivated by coding in distributed storage, we attempt to repair a failed node by accessing information stored on d helper nodes, where k ≼ d ≼ n – 1, and using as little repair bandwidth as possible to recover the lost information.By the so-called cut-set bound (Dimakis et al., 2010), the repair bandwidth of an (n,k = n – r) MDS code using d helper nodes is at least dl/(d + 1 – k), where l is the size of the node. A number of constructions of MDS array codes have been shown to meet this bound with equality. In a related but separate line of work, Guruswami and Wootters (2016) studied repair of Reed-Solomon (RS) codes, showing that it is possible to perform repair using a smaller bandwidth than under the trivial approach. At the same time, their work as well as follow-up papers stopped short of constructing RS codes (or any scalar MDS codes) that meet the cut-set bound with equality, which has been an open problem in coding theory.In this work we present a solution to this problem, constructing RS codes of length n over the field of size (ql, l = exp((1 + o(1)n log n) that meet the cut-set bound. We also prove an almost matching lower bound on l, showing that super-exponential scaling is both necessary and sufficient for achieving the cut-set bound using linear repair schemes. More precisely, we prove that for scalar MDS codes (including the RS codes) to meet this bound, the sub-packetization l must satisfy l ≽ exp((1 + o(1))k log k).","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"14 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122362228","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 if the nearly-linear time solvers for Laplacian matrices and their generalizations can be extended to solve just slightly larger families of linear systems, then they can be used to quickly solve all systems of linear equations over the reals. This result can be viewed either positively or negatively: either we will develop nearly-linear time algorithms for solving all systems of linear equations over the reals, or progress on the families we can solve in nearly-linear time will soon halt.
{"title":"Hardness Results for Structured Linear Systems","authors":"Rasmus Kyng, Peng Zhang","doi":"10.1109/FOCS.2017.69","DOIUrl":"https://doi.org/10.1109/FOCS.2017.69","url":null,"abstract":"We show that if the nearly-linear time solvers for Laplacian matrices and their generalizations can be extended to solve just slightly larger families of linear systems, then they can be used to quickly solve all systems of linear equations over the reals. This result can be viewed either positively or negatively: either we will develop nearly-linear time algorithms for solving all systems of linear equations over the reals, or progress on the families we can solve in nearly-linear time will soon halt.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"11 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121497646","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}
Pub Date : 2017-05-04DOI: 10.22331/q-2020-02-14-230
Joran van Apeldoorn, A. Gilyén, S. Gribling, R. D. Wolf
Brandão and Svore recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension n of the problem and the number m of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure.We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with mn when m is approximately n, which is the same as classical.
{"title":"Quantum SDP-Solvers: Better Upper and Lower Bounds","authors":"Joran van Apeldoorn, A. Gilyén, S. Gribling, R. D. Wolf","doi":"10.22331/q-2020-02-14-230","DOIUrl":"https://doi.org/10.22331/q-2020-02-14-230","url":null,"abstract":"Brandão and Svore recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension n of the problem and the number m of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure.We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with mn when m is approximately n, which is the same as classical.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133794428","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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