We present a deterministic distributed algorithm that computes a (2δ-1)-edge-coloring, or even list-edge-coloring, in any n-node graph with maximum degree δ, in O(log^8 δ ⋅ log n) rounds. This answers one of the long-standing open questions of distributed graph algorithms} from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities were 2^{O(√{log n})} by Panconesi and Srinivasan [STOC92] and Õ(√{δ}) + O(log^* n) by Fraigniaud, Heinrich, and Kosowski [FOCS16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of (2δ-1)-edge-coloring to poly(loglog n) rounds.The key technical ingredient is a deterministic distributed algorithm for hypergraph maximal matching, which we believe will be of interest beyond this result. In any hypergraph of rank r — where each hyperedge has at most r vertices — with n nodes and maximum degree δ, this algorithm computes a maximal matching in O(r^5 log^{6+log r } δ ⋅ log n) rounds.This hypergraph matching algorithm and its extensions also lead to a number of other results. In particular, we obtain a polylogarithmic-time deterministic distributed maximal independent set (MIS) algorithm for graphs with bounded neighborhood independence, hence answering Open Problem 5 of Barenboim and Elkins book, a big((log δ/ε)^{O(log 1/ε)}big)-round deterministic algorithm for (1+ε)-approximation of maximum matching, and a quasi-polylogarithmic-time deterministic distributed algorithm for orienting λ-arboricity graphs with out-degree at most lceil (1+ε)λ rceil, for any constant ε 0, hence partially answering Open Problem 10 of Barenboim and Elkins book.
{"title":"Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching","authors":"Manuela Fischer, M. Ghaffari, F. Kuhn","doi":"10.1109/FOCS.2017.25","DOIUrl":"https://doi.org/10.1109/FOCS.2017.25","url":null,"abstract":"We present a deterministic distributed algorithm that computes a (2δ-1)-edge-coloring, or even list-edge-coloring, in any n-node graph with maximum degree δ, in O(log^8 δ ⋅ log n) rounds. This answers one of the long-standing open questions of distributed graph algorithms} from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities were 2^{O(√{log n})} by Panconesi and Srinivasan [STOC92] and Õ(√{δ}) + O(log^* n) by Fraigniaud, Heinrich, and Kosowski [FOCS16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of (2δ-1)-edge-coloring to poly(loglog n) rounds.The key technical ingredient is a deterministic distributed algorithm for hypergraph maximal matching, which we believe will be of interest beyond this result. In any hypergraph of rank r — where each hyperedge has at most r vertices — with n nodes and maximum degree δ, this algorithm computes a maximal matching in O(r^5 log^{6+log r } δ ⋅ log n) rounds.This hypergraph matching algorithm and its extensions also lead to a number of other results. In particular, we obtain a polylogarithmic-time deterministic distributed maximal independent set (MIS) algorithm for graphs with bounded neighborhood independence, hence answering Open Problem 5 of Barenboim and Elkins book, a big((log δ/ε)^{O(log 1/ε)}big)-round deterministic algorithm for (1+ε)-approximation of maximum matching, and a quasi-polylogarithmic-time deterministic distributed algorithm for orienting λ-arboricity graphs with out-degree at most lceil (1+ε)λ rceil, for any constant ε 0, hence partially answering Open Problem 10 of Barenboim and Elkins book.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129716551","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}
Michael B. Cohen, A. Madry, Dimitris Tsipras, Adrian Vladu
In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time widetilde{O}(mlog kappa log^2 (1/≥ilon)) where ≥ilon is the amount of error we are willing to tolerate. Here, kappa represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever kappa is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time widetilde{O}(m^{3/2} log (1/≥ilon)).In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient.
{"title":"Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods","authors":"Michael B. Cohen, A. Madry, Dimitris Tsipras, Adrian Vladu","doi":"10.1109/FOCS.2017.88","DOIUrl":"https://doi.org/10.1109/FOCS.2017.88","url":null,"abstract":"In this paper, we study matrix scaling and balancing, which are fundamental problems in scientific computing, with a long line of work on them that dates back to the 1960s. We provide algorithms for both these problems that, ignoring logarithmic factors involving the dimension of the input matrix and the size of its entries, both run in time widetilde{O}(mlog kappa log^2 (1/≥ilon)) where ≥ilon is the amount of error we are willing to tolerate. Here, kappa represents the ratio between the largest and the smallest entries of the optimal scalings. This implies that our algorithms run in nearly-linear time whenever kappa is quasi-polynomial, which includes, in particular, the case of strictly positive matrices. We complement our results by providing a separate algorithm that uses an interior-point method and runs in time widetilde{O}(m^{3/2} log (1/≥ilon)).In order to establish these results, we develop a new second-order optimization framework that enables us to treat both problems in a unified and principled manner. This framework identifies a certain generalization of linear system solving that we can use to efficiently minimize a broad class of functions, which we call second-order robust. We then show that in the context of the specific functions capturing matrix scaling and balancing, we can leverage and generalize the work on Laplacian system solving to make the algorithms obtained via this framework very efficient.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128138369","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}
Zeyuan Allen-Zhu, Yuanzhi Li, R. Oliveira, A. Wigderson
We develop several efficient algorithms for the classical Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input n× n matrix A, this problem asks to find diagonal (scaling) matrices X and Y (if they exist), so that X A Y ε-approximates a doubly stochastic matrix, or more generally a matrix with prescribed row and column sums.We address the general scaling problem as well as some important special cases. In particular, if A has m nonzero entries, and if there exist X and Y with polynomially large entries such that X A Y is doubly stochastic, then we can solve the problem in total complexity tilde{O}(m + n^{4/3}). This greatly improves on the best known previous results, which were either tilde{O}(n^4) or O(m n^{1/2}/ε).Our algorithms are based on tailor-made first and second order techniques, combined with other recent advances in continuous optimization, which may be of independent interest for solving similar problems.
我们为经典的矩阵缩放问题开发了几种有效的算法，这些算法被用于许多不同的领域，从预处理线性系统到近似永久系统。输入n×在矩阵A中，这个问题要求找到对角(缩放)矩阵X和Y(如果它们存在)，使得X A Y ε-近似于一个双重随机矩阵，或者更一般地说，一个具有规定的行和和的矩阵。我们讨论了一般的标度问题以及一些重要的特殊情况。特别地，如果A有m个非零项，并且如果X和Y有多项式大的项，使得X A Y是双随机的，那么我们可以用总复杂度tilde{O}(m + n^{4/3})来解决问题。这大大改进了以前最著名的结果，即tilde{O}(n^4)或O(m n^{1/2}/ε)。我们的算法基于定制的一阶和二阶技术，结合了其他最近在连续优化方面的进展，这可能对解决类似问题有独立的兴趣。
{"title":"Much Faster Algorithms for Matrix Scaling","authors":"Zeyuan Allen-Zhu, Yuanzhi Li, R. Oliveira, A. Wigderson","doi":"10.1109/FOCS.2017.87","DOIUrl":"https://doi.org/10.1109/FOCS.2017.87","url":null,"abstract":"We develop several efficient algorithms for the classical Matrix Scaling} problem, which is used in many diverse areas, from preconditioning linear systems to approximation of the permanent. On an input n× n matrix A, this problem asks to find diagonal (scaling) matrices X and Y (if they exist), so that X A Y ε-approximates a doubly stochastic matrix, or more generally a matrix with prescribed row and column sums.We address the general scaling problem as well as some important special cases. In particular, if A has m nonzero entries, and if there exist X and Y with polynomially large entries such that X A Y is doubly stochastic, then we can solve the problem in total complexity tilde{O}(m + n^{4/3}). This greatly improves on the best known previous results, which were either tilde{O}(n^4) or O(m n^{1/2}/ε).Our algorithms are based on tailor-made first and second order techniques, combined with other recent advances in continuous optimization, which may be of independent interest for solving similar problems.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"343 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116531755","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 properties of edge-colored vertex-ordered graphs} – graphs with a totally ordered vertex set and a finite set of possible edge colors – showing that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries. We also explain how the proof can be adapted to show that any hereditary property of two-dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable. The first result generalizes the result of Alon and Shapira [FOCS05; SICOMP08], who showed that any hereditary graph property (without vertex order) is strongly testable. The second result answers and generalizes a conjecture of Alon, Fischer and Newman [SICOMP07] concerning testing of matrix properties.The testability is proved by establishing a removal lemma for vertex-ordered graphs. It states that if such a graph is far enough from satisfying a certain hereditary property, then most of its induced vertex-ordered subgraphs on a certain (large enough) constant number of vertices do not satisfy the property as well.The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques. Along the way we develop a Ramsey-type lemma for multipartite graphs with undesirable edges, stating that one can find a Ramsey-type structure in such a graph, in which the density of the undesirable edges is not much higher than the density of those edges in the graph.
{"title":"Testing Hereditary Properties of Ordered Graphs and Matrices","authors":"N. Alon, Omri Ben-Eliezer, E. Fischer","doi":"10.1109/FOCS.2017.83","DOIUrl":"https://doi.org/10.1109/FOCS.2017.83","url":null,"abstract":"We consider properties of edge-colored vertex-ordered graphs} – graphs with a totally ordered vertex set and a finite set of possible edge colors – showing that any hereditary property of such graphs is strongly testable, i.e., testable with a constant number of queries. We also explain how the proof can be adapted to show that any hereditary property of two-dimensional matrices over a finite alphabet (where row and column order is not ignored) is strongly testable. The first result generalizes the result of Alon and Shapira [FOCS05; SICOMP08], who showed that any hereditary graph property (without vertex order) is strongly testable. The second result answers and generalizes a conjecture of Alon, Fischer and Newman [SICOMP07] concerning testing of matrix properties.The testability is proved by establishing a removal lemma for vertex-ordered graphs. It states that if such a graph is far enough from satisfying a certain hereditary property, then most of its induced vertex-ordered subgraphs on a certain (large enough) constant number of vertices do not satisfy the property as well.The proof bridges the gap between techniques related to the regularity lemma, used in the long chain of papers investigating graph testing, and string testing techniques. Along the way we develop a Ramsey-type lemma for multipartite graphs with undesirable edges, stating that one can find a Ramsey-type structure in such a graph, in which the density of the undesirable edges is not much higher than the density of those edges in the graph.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131297044","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}
Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. The standard way to parametrize interesting subclasses of the constraint satisfaction problem is via finite constraint languages. The main problem is to classify those subclasses that are solvable in polynomial time and those that are NP-complete. It was conjectured that if a core of a constraint language has a weak near unanimity polymorphism then the corresponding constraint satisfaction problem is tractable, otherwise it is NP-complete.In the paper we present an algorithm that solves Constraint Satisfaction Problem in polynomial time for constraint languages having a weak near unanimity polymorphism, which proves the remaining part of the conjecture.
{"title":"A Proof of CSP Dichotomy Conjecture","authors":"Dmitriy Zhuk","doi":"10.1109/FOCS.2017.38","DOIUrl":"https://doi.org/10.1109/FOCS.2017.38","url":null,"abstract":"Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. The standard way to parametrize interesting subclasses of the constraint satisfaction problem is via finite constraint languages. The main problem is to classify those subclasses that are solvable in polynomial time and those that are NP-complete. It was conjectured that if a core of a constraint language has a weak near unanimity polymorphism then the corresponding constraint satisfaction problem is tractable, otherwise it is NP-complete.In the paper we present an algorithm that solves Constraint Satisfaction Problem in polynomial time for constraint languages having a weak near unanimity polymorphism, which proves the remaining part of the conjecture.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"18 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131004843","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 approximate similarity (near neighbour) search can be solved in high dimensions with performance matching state of the art (data independent) Locality Sensitive Hashing, but with a guarantee of no false negatives. Specifically we give two data structures for common problems. For c-approximate near neighbour in Hamming space, for which we get query time dn^{1/c+o(1)} and space dn^{1+1/c+o(1)} matching that of [Indyk and Motwani, 1998] and answering a long standing open question from [Indyk, 2000a] and [Pagh, 2016] in the affirmative. For (s1, s2)-approximate Jaccard similarity we get query time d^2n^{ρ+o(1)} and space d^2n^{1+ρ+o(1), ρ= [log (1+s1)/(2s1)]/[log (1+s2)/(2s2)], when sets have equal size, matching the performance of [Pagh and Christiani, 2017].We use space partitions as in classic LSH, but construct these using a combination of brute force, tensoring and splitter functions à la [Naor et al., 1995]. We also show two dimensionality reduction lemmas with 1-sided error.
我们表明，近似相似性(近邻)搜索可以在高维中解决，性能匹配最新的(数据独立的)局部敏感哈希，但保证没有假阴性。具体来说，我们给出了两种常见问题的数据结构。对于Hamming空间中的c-近似近邻，我们得到了与[Indyk and Motwani, 1998]匹配的查询时间dn^{1/c+o(1)}和空间dn^{1+1/c+o(1)}，肯定地回答了[Indyk, 2000a]和[Pagh, 2016]中一个长期存在的开放问题。对于(s1, s2)-近似Jaccard相似度，当集合大小相等时，我们得到查询时间d^2n^{ρ+o(1)}和空间d^2n^{1+ρ+o(1)， ρ= [log (1+s1)/(2s1)]/[log (1+s2)/(2s2)]，性能与[Pagh and Christiani, 2017]相匹配。我们像在经典的LSH中一样使用空间分区，但是使用蛮力、张紧和分割函数的组合来构建它们à[Naor et al.， 1995]。我们还展示了具有单边误差的两个降维引理。
{"title":"Optimal Las Vegas Locality Sensitive Data Structures","authors":"Thomas Dybdahl Ahle","doi":"10.1109/FOCS.2017.91","DOIUrl":"https://doi.org/10.1109/FOCS.2017.91","url":null,"abstract":"We show that approximate similarity (near neighbour) search can be solved in high dimensions with performance matching state of the art (data independent) Locality Sensitive Hashing, but with a guarantee of no false negatives. Specifically we give two data structures for common problems. For c-approximate near neighbour in Hamming space, for which we get query time dn^{1/c+o(1)} and space dn^{1+1/c+o(1)} matching that of [Indyk and Motwani, 1998] and answering a long standing open question from [Indyk, 2000a] and [Pagh, 2016] in the affirmative. For (s1, s2)-approximate Jaccard similarity we get query time d^2n^{ρ+o(1)} and space d^2n^{1+ρ+o(1), ρ= [log (1+s1)/(2s1)]/[log (1+s2)/(2s2)], when sets have equal size, matching the performance of [Pagh and Christiani, 2017].We use space partitions as in classic LSH, but construct these using a combination of brute force, tensoring and splitter functions à la [Naor et al., 1995]. We also show two dimensionality reduction lemmas with 1-sided error.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125245645","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 the perfect matching problem in general graphs is in Quasi-NC. That is, we give a deterministic parallel algorithm which runs in O(log^3 n) time on n^{O(log^2 n)} processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to obtain a Randomized NC algorithm.Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.
{"title":"The Matching Problem in General Graphs Is in Quasi-NC","authors":"O. Svensson, Jakub Tarnawski","doi":"10.1109/FOCS.2017.70","DOIUrl":"https://doi.org/10.1109/FOCS.2017.70","url":null,"abstract":"We show that the perfect matching problem in general graphs is in Quasi-NC. That is, we give a deterministic parallel algorithm which runs in O(log^3 n) time on n^{O(log^2 n)} processors. The result is obtained by a derandomization of the Isolation Lemma for perfect matchings, which was introduced in the classic paper by Mulmuley, Vazirani and Vazirani [1987] to obtain a Randomized NC algorithm.Our proof extends the framework of Fenner, Gurjar and Thierauf [2016], who proved the analogous result in the special case of bipartite graphs. Compared to that setting, several new ingredients are needed due to the significantly more complex structure of perfect matchings in general graphs. In particular, our proof heavily relies on the laminar structure of the faces of the perfect matching polytope.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115628838","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. Kapralov, Jelani Nelson, J. Pachocki, Zhengyu Wang, David P. Woodruff, Mobin Yahyazadeh
In the communication problem UR (universal relation), Alice and Bob respectively receive x, y ∊{0,1}^n with the promise that x≠ y. The last player to receive a message must output an index i such that x_i≠ y_i. We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly Theta(min{n,log(1/δ)log^2(frac n{log(1/δ)})}) for failure probability δ. Our lower bound holds even if promised mathop{support}(y)⊄ mathop{support}(x). As a corollary, we obtain optimal lower bounds for ℓ_p-sampling in strict turnstile streams for 0le p streams for 0 ≤ p
在通信问题UR (universal relation)中，Alice和Bob分别收到x, y ∊{0,1}^n，并承诺x≠y.最后一个接收到消息的播放器必须输出索引i，这样x_i≠y_i。我们证明了该问题在公共币模型中的随机单向通信复杂度正好是Theta(min{n，log(1/δ)log^2(frac n{log(1/δ)})})对于失败概率δ。我们的下限保持不变，即使承诺mathop{support}(y)⊄ mathop{支持}(x)。作为一个结论，我们得到了严格旋转门流中_# x2113;_p采样的最优下界，为0 p流为0 ≤p
{"title":"Optimal Lower Bounds for Universal Relation, and for Samplers and Finding Duplicates in Streams","authors":"M. Kapralov, Jelani Nelson, J. Pachocki, Zhengyu Wang, David P. Woodruff, Mobin Yahyazadeh","doi":"10.1109/FOCS.2017.50","DOIUrl":"https://doi.org/10.1109/FOCS.2017.50","url":null,"abstract":"In the communication problem UR (universal relation), Alice and Bob respectively receive x, y ∊{0,1}^n with the promise that x≠ y. The last player to receive a message must output an index i such that x_i≠ y_i. We prove that the randomized one-way communication complexity of this problem in the public coin model is exactly Theta(min{n,log(1/δ)log^2(frac n{log(1/δ)})}) for failure probability δ. Our lower bound holds even if promised mathop{support}(y)⊄ mathop{support}(x). As a corollary, we obtain optimal lower bounds for ℓ_p-sampling in strict turnstile streams for 0le p streams for 0 ≤ p","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122417310","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 approximate degree of a Boolean function f: {-1, 1}^n ↦ {-1, 1} is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits.Specifically, we show how to transform any Boolean function f with approximate degree d into a function F on O(n polylog(n)) variables with approximate degree at least D = Ω(n^{1/3} d^{2/3}). In particular, if d = n^{1-Ω(1), then D is polynomially larger than d. Moreover, if f is computed by a constant-depth polynomial-size Boolean circuit, then so is F.By recursively applying our transformation, for any constant δ 0 we exhibit an AC° function of approximate degree Ω(n^{1-δ}). This improves over the best previous lower bound of Ω(n^{2/3}) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of n that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width.We describe several applications of these results. We give:• For any constant δ 0, an Ω(n^{1-δ}) lower bound on the quantum communication complexity of a function in AC°.• A Boolean function f with approximate degree at least C(f)^{2-o(1), where C(f) is the certificate complexity of f. This separation is optimal up to the o(1) term in the exponent.• Improved secret sharing schemes with reconstruction procedures in AC°.
{"title":"A Nearly Optimal Lower Bound on the Approximate Degree of AC^0","authors":"Mark Bun, J. Thaler","doi":"10.1109/FOCS.2017.10","DOIUrl":"https://doi.org/10.1109/FOCS.2017.10","url":null,"abstract":"The approximate degree of a Boolean function f: {-1, 1}^n ↦ {-1, 1} is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits.Specifically, we show how to transform any Boolean function f with approximate degree d into a function F on O(n polylog(n)) variables with approximate degree at least D = Ω(n^{1/3} d^{2/3}). In particular, if d = n^{1-Ω(1), then D is polynomially larger than d. Moreover, if f is computed by a constant-depth polynomial-size Boolean circuit, then so is F.By recursively applying our transformation, for any constant δ 0 we exhibit an AC° function of approximate degree Ω(n^{1-δ}). This improves over the best previous lower bound of Ω(n^{2/3}) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of n that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width.We describe several applications of these results. We give:• For any constant δ 0, an Ω(n^{1-δ}) lower bound on the quantum communication complexity of a function in AC°.• A Boolean function f with approximate degree at least C(f)^{2-o(1), where C(f) is the certificate complexity of f. This separation is optimal up to the o(1) term in the exponent.• Improved secret sharing schemes with reconstruction procedures in AC°.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130168218","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 a non-uniform Constraint Satisfaction problem CSP(Γ), where G is a set of relations on a finite set A, the goal is to find an assignment of values to variables subject to constraints imposed on specified sets of variables using the relations from Γ. The Dichotomy Conjecture for the non-uniform CSP states that for every constraint language Gm the problem CSP(Γ) is either solvable in polynomial time or is NP-complete. It was proposed by Feder and Vardi in their seminal 1993 paper. In this paper we confirm the Dichotomy Conjecture.
{"title":"A Dichotomy Theorem for Nonuniform CSPs","authors":"A. Bulatov","doi":"10.1109/FOCS.2017.37","DOIUrl":"https://doi.org/10.1109/FOCS.2017.37","url":null,"abstract":"In a non-uniform Constraint Satisfaction problem CSP(Γ), where G is a set of relations on a finite set A, the goal is to find an assignment of values to variables subject to constraints imposed on specified sets of variables using the relations from Γ. The Dichotomy Conjecture for the non-uniform CSP states that for every constraint language Gm the problem CSP(Γ) is either solvable in polynomial time or is NP-complete. It was proposed by Feder and Vardi in their seminal 1993 paper. In this paper we confirm the Dichotomy Conjecture.","PeriodicalId":311592,"journal":{"name":"2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124386174","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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