An approximate membership data structure is a randomized data structure for representing a set which supports membership queries. It allows for a small false positive error rate but has no false negative errors. Such data structures were first introduced by Bloom in the 1970's, and have since had numerous applications, mainly in distributed systems, database systems, and networks. The algorithm of Bloom is quite effective: it can store a set $S$ of size $n$ by using only $approx 1.44 n log_2(1/epsilon)$ bits while having false positive error $epsilon$. This is within a constant factor of the entropy lower bound of $n log_2(1/epsilon)$ for storing such sets. Closing this gap is an important open problem, as Bloom filters are widely used is situations were storage is at a premium. Bloom filters have another property: they are dynamic. That is, they support the iterative insertions of up to $n$ elements. In fact, if one removes this requirement, there exist static data structures which receive the entire set at once and can almost achieve the entropy lower bound, they require only $n log_2(1/epsilon)(1+o(1))$ bits. Our main result is a new lower bound for the memory requirements of any dynamic approximate membership data structure. We show that for any constant $epsilon>0$, any such data structure which achieves false positive error rate of $epsilon$ must use at least $C(epsilon) cdot n log_2(1/epsilon)$ memory bits, where $C(epsilon)>1$ depends only on $epsilon$. This shows that the entropy lower bound cannot be achieved by dynamic data structures for any constant error rate. In fact, our lower bound holds even in the setting where the insertion and query algorithms may use shared randomness, and where they are only required to perform well on average.
近似成员关系数据结构是一种随机数据结构，用于表示支持成员关系查询的集合。它允许一个小的假阳性错误率，但没有假阴性错误。这样的数据结构最初是由Bloom在20世纪70年代引入的，并且已经有了大量的应用，主要是在分布式系统、数据库系统和网络中。Bloom算法是非常有效的:它可以存储一个$S$大小的集合$n$，只使用$approx 1.44 n log_2(1/epsilon)$位，同时有误报错误$epsilon$。这在存储这样的集合的熵下界$n log_2(1/epsilon)$的一个常数因子之内。缩小这一差距是一个重要的开放问题，因为布隆过滤器在存储非常宝贵的情况下被广泛使用。布隆过滤器还有另一个属性:它们是动态的。也就是说，它们支持最多$n$元素的迭代插入。事实上，如果去掉这个要求，存在静态数据结构，它可以一次接收整个集合，几乎可以达到熵的下界，它们只需要$n log_2(1/epsilon)(1+o(1))$位。我们的主要结果是任何动态近似隶属度数据结构的内存需求的一个新的下界。我们表明，对于任何常数$epsilon>0$，任何达到假阳性错误率$epsilon$的此类数据结构必须至少使用$C(epsilon) cdot n log_2(1/epsilon)$内存位，其中$C(epsilon)>1$仅依赖于$epsilon$。这表明，对于任意恒定的错误率，动态数据结构无法达到熵的下界。事实上，我们的下界甚至适用于插入和查询算法可能使用共享随机性的情况，以及它们只需要平均表现良好的情况。
{"title":"A Lower Bound for Dynamic Approximate Membership Data Structures","authors":"Shachar Lovett, E. Porat","doi":"10.1109/FOCS.2010.81","DOIUrl":"https://doi.org/10.1109/FOCS.2010.81","url":null,"abstract":"An approximate membership data structure is a randomized data structure for representing a set which supports membership queries. It allows for a small false positive error rate but has no false negative errors. Such data structures were first introduced by Bloom in the 1970's, and have since had numerous applications, mainly in distributed systems, database systems, and networks. The algorithm of Bloom is quite effective: it can store a set $S$ of size $n$ by using only $approx 1.44 n log_2(1/epsilon)$ bits while having false positive error $epsilon$. This is within a constant factor of the entropy lower bound of $n log_2(1/epsilon)$ for storing such sets. Closing this gap is an important open problem, as Bloom filters are widely used is situations were storage is at a premium. Bloom filters have another property: they are dynamic. That is, they support the iterative insertions of up to $n$ elements. In fact, if one removes this requirement, there exist static data structures which receive the entire set at once and can almost achieve the entropy lower bound, they require only $n log_2(1/epsilon)(1+o(1))$ bits. Our main result is a new lower bound for the memory requirements of any dynamic approximate membership data structure. We show that for any constant $epsilon>0$, any such data structure which achieves false positive error rate of $epsilon$ must use at least $C(epsilon) cdot n log_2(1/epsilon)$ memory bits, where $C(epsilon)>1$ depends only on $epsilon$. This shows that the entropy lower bound cannot be achieved by dynamic data structures for any constant error rate. In fact, our lower bound holds even in the setting where the insertion and query algorithms may use shared randomness, and where they are only required to perform well on average.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"50 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134330821","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 the following general scheduling problem: The input consists of $n$ jobs, each with an arbitrary release time, size, and a monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as weighted flow, weighted tardiness, and sum of flow squared. The main contribution of this paper is a randomized polynomial-time algorithm with an approximation ratio $O(log log nP )$, where $P$ is the maximum job size. We also give an $O(1)$ approximation in the special case when all jobs have identical release times. Initially, we show how to reduce this scheduling problem to a particular geometric set-cover problem. We then consider a natural linear programming formulation of this geometric set-cover problem, strengthened by adding knapsack cover inequalities, and show that rounding the solution of this linear program can be reduced to other particular geometric set-cover problems. We then develop algorithms for these sub-problems using the local ratio technique, and Varadarajan's quasi-uniform sampling technique. This general algorithmic approach improves the best known approximation ratios by at least an exponential factor (and much more in some cases) for essentially all of the nontrivial common special cases of this problem. We believe that this geometric interpretation of scheduling is of independent interest.
{"title":"The Geometry of Scheduling","authors":"N. Bansal, K. Pruhs","doi":"10.1137/130911317","DOIUrl":"https://doi.org/10.1137/130911317","url":null,"abstract":"We consider the following general scheduling problem: The input consists of $n$ jobs, each with an arbitrary release time, size, and a monotone function specifying the cost incurred when the job is completed at a particular time. The objective is to find a preemptive schedule of minimum aggregate cost. This problem formulation is general enough to include many natural scheduling objectives, such as weighted flow, weighted tardiness, and sum of flow squared. The main contribution of this paper is a randomized polynomial-time algorithm with an approximation ratio $O(log log nP )$, where $P$ is the maximum job size. We also give an $O(1)$ approximation in the special case when all jobs have identical release times. Initially, we show how to reduce this scheduling problem to a particular geometric set-cover problem. We then consider a natural linear programming formulation of this geometric set-cover problem, strengthened by adding knapsack cover inequalities, and show that rounding the solution of this linear program can be reduced to other particular geometric set-cover problems. We then develop algorithms for these sub-problems using the local ratio technique, and Varadarajan's quasi-uniform sampling technique. This general algorithmic approach improves the best known approximation ratios by at least an exponential factor (and much more in some cases) for essentially all of the nontrivial common special cases of this problem. We believe that this geometric interpretation of scheduling is of independent interest.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126917064","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}
Much of the literature on rational cryptography focuses on analyzing the strategic properties of cryptographic protocols. However, due to the presence of computationally-bounded players and the asymptotic nature of cryptographic security, a definition of sequential rationality for this setting has thus far eluded researchers. We propose a new framework for overcoming these obstacles, and provide the first definitions of computational solution concepts that guarantee sequential rationality. We argue that natural computational variants of sub game perfection are too strong for cryptographic protocols. As an alternative, we introduce a weakening called threat-free Nash equilibrium that is more permissive but still eliminates the undesirable ``empty threats'' of non-sequential solution concepts. To demonstrate the applicability of our framework, we revisit the problem of implementing a mediator for correlated equilibria (Dodis-Halevi-Rabin, Crypto'00), and propose a variant of their protocol that is sequentially rational for a non-trivial class of correlated equilibria. Our treatment provides a better understanding of the conditions under which mediators in a correlated equilibrium can be replaced by a stable protocol.
{"title":"Sequential Rationality in Cryptographic Protocols","authors":"R. Gradwohl, N. Livne, Alon Rosen","doi":"10.1145/2399187.2399189","DOIUrl":"https://doi.org/10.1145/2399187.2399189","url":null,"abstract":"Much of the literature on rational cryptography focuses on analyzing the strategic properties of cryptographic protocols. However, due to the presence of computationally-bounded players and the asymptotic nature of cryptographic security, a definition of sequential rationality for this setting has thus far eluded researchers. We propose a new framework for overcoming these obstacles, and provide the first definitions of computational solution concepts that guarantee sequential rationality. We argue that natural computational variants of sub game perfection are too strong for cryptographic protocols. As an alternative, we introduce a weakening called threat-free Nash equilibrium that is more permissive but still eliminates the undesirable ``empty threats'' of non-sequential solution concepts. To demonstrate the applicability of our framework, we revisit the problem of implementing a mediator for correlated equilibria (Dodis-Halevi-Rabin, Crypto'00), and propose a variant of their protocol that is sequentially rational for a non-trivial class of correlated equilibria. Our treatment provides a better understanding of the conditions under which mediators in a correlated equilibrium can be replaced by a stable protocol.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131186853","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 general method of designing fast approximation algorithms for cut-based minimization problems in undirected graphs. In particular, we develop a technique that given any such problem that can be approximated quickly on trees, allows approximating it almost as quickly on general graphs while only losing a poly-logarithmic factor in the approximation guarantee. To illustrate the applicability of our paradigm, we focus our attention on the undirected sparsest cut problem with general demands and the balanced separator problem. By a simple use of our framework, we obtain poly-logarithmic approximation algorithms for these problems that run in time close to linear. The main tool behind our result is an efficient procedure that decomposes general graphs into simpler ones while approximately preserving the cut-flow structure. This decomposition is inspired by the cut-based graph decomposition of R"acke that was developed in the context of oblivious routing schemes, as well as, by the construction of the ultrasparsifiers due to Spiel man and Teng that was employed to preconditioning symmetric diagonally-dominant matrices.
{"title":"Fast Approximation Algorithms for Cut-Based Problems in Undirected Graphs","authors":"A. Madry","doi":"10.1109/FOCS.2010.30","DOIUrl":"https://doi.org/10.1109/FOCS.2010.30","url":null,"abstract":"We present a general method of designing fast approximation algorithms for cut-based minimization problems in undirected graphs. In particular, we develop a technique that given any such problem that can be approximated quickly on trees, allows approximating it almost as quickly on general graphs while only losing a poly-logarithmic factor in the approximation guarantee. To illustrate the applicability of our paradigm, we focus our attention on the undirected sparsest cut problem with general demands and the balanced separator problem. By a simple use of our framework, we obtain poly-logarithmic approximation algorithms for these problems that run in time close to linear. The main tool behind our result is an efficient procedure that decomposes general graphs into simpler ones while approximately preserving the cut-flow structure. This decomposition is inspired by the cut-based graph decomposition of R\"acke that was developed in the context of oblivious routing schemes, as well as, by the construction of the ultrasparsifiers due to Spiel man and Teng that was employed to preconditioning symmetric diagonally-dominant matrices.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"84 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116448847","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 Monte Carlo algorithm for Hamilton city detection in an $n$-vertex undirected graph running in $O^*(1.657^{n})$ time. To the best of our knowledge, this is the first super polynomial improvement on the worst case runtime for the problem since the $O^*(2^n)$ bound established for TSP almost fifty years ago (Bellman 1962, Held and Karp 1962). It answers in part the first open problem in Woe ginger's 2003 survey on exact algorithms for NP-hard problems. For bipartite graphs, we improve the bound to $O^*(1.414^{n})$ time. Both the bipartite and the general algorithm can be implemented to use space polynomial in $n$. We combine several recently resurrected ideas to get the results. Our main technical contribution is a new reduction inspired by the algebraic sieving method for $k$-Path (Koutis ICALP 2008, Williams IPL 2009). We introduce the Labeled Cycle Cover Sum in which we are set to count weighted arc labeled cycle covers over a finite field of characteristic two. We reduce Hamiltonicity to Labeled Cycle Cover Sum and apply the determinant summation technique for Exact Set Covers (Bj"orklund STACS 2010) to evaluate it.
在$O^*(1.657^{n})$时间运行的$n$顶点无向图中，提出了一种用于Hamilton城市检测的蒙特卡罗算法。据我们所知，这是自50年前为TSP建立的$O^*(2^n)$界以来，该问题在最坏情况下的第一个超级多项式改进(Bellman 1962, Held and Karp 1962)。它部分地回答了2003年Woe ginger关于np困难问题的精确算法的调查中的第一个开放问题。对于二部图，我们将边界改进为$O^*(1.414^{n})$时间。二部算法和一般算法都可以使用空间多项式来实现。我们结合了几个最近复活的想法来得到结果。我们的主要技术贡献是受k -Path的代数筛分方法启发的一种新的简化(Koutis ICALP 2008, Williams IPL 2009)。我们引入了标记循环覆盖和，其中我们被设置为在特征为2的有限域上计数加权弧标记循环覆盖。我们将哈密性约简为标记循环覆盖和，并应用精确集覆盖的行列式求和技术(Bj orklund STACS 2010)来评估它。
{"title":"Determinant Sums for Undirected Hamiltonicity","authors":"Andreas Björklund","doi":"10.1137/110839229","DOIUrl":"https://doi.org/10.1137/110839229","url":null,"abstract":"We present a Monte Carlo algorithm for Hamilton city detection in an $n$-vertex undirected graph running in $O^*(1.657^{n})$ time. To the best of our knowledge, this is the first super polynomial improvement on the worst case runtime for the problem since the $O^*(2^n)$ bound established for TSP almost fifty years ago (Bellman 1962, Held and Karp 1962). It answers in part the first open problem in Woe ginger's 2003 survey on exact algorithms for NP-hard problems. For bipartite graphs, we improve the bound to $O^*(1.414^{n})$ time. Both the bipartite and the general algorithm can be implemented to use space polynomial in $n$. We combine several recently resurrected ideas to get the results. Our main technical contribution is a new reduction inspired by the algebraic sieving method for $k$-Path (Koutis ICALP 2008, Williams IPL 2009). We introduce the Labeled Cycle Cover Sum in which we are set to count weighted arc labeled cycle covers over a finite field of characteristic two. We reduce Hamiltonicity to Labeled Cycle Cover Sum and apply the determinant summation technique for Exact Set Covers (Bj\"orklund STACS 2010) to evaluate it.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128543720","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}
Valiant introduced match gate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, match gate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in statistical physics community for decades. They capture precisely those problems which are #P-hard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs, and can be arbitrary real-valued symmetric functions. We prove that, every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are #P-hard on general graphs but ractable on planar graphs, or (3) those which are #P-hard even on planar graphs. The classification criteria are explicit. Moreover, problems in category (2) are tractable on planar graphs precisely by holographic algorithms with matchgates.
Valiant介绍了匹配门计算和全息算法。许多看似指数时间的问题都可以用多项式时间来解决。我们表明，在很强的意义上，匹配门计算和基于它们的全息算法为统计物理学界几十年来研究的广泛的计数问题提供了一种通用的方法。它们精确地捕获了那些在一般图上#P-hard但在平面图上可在多项式时间内计算的问题。更准确地说，我们在计数CSP问题的框架下证明了复杂度二分定理。局部约束函数采用布尔输入，可以是任意实值对称函数。我们证明了该类中的每一个问题都精确地属于三类:(1)一般图上可处理(即多项式时间可计算)的问题，或(2)一般图上# p -难但在平面图上可处理的问题，或(3)即使在平面图上# p -难的问题。分类标准是明确的。此外，用带匹配门的全息算法可以精确地处理平面图上的第(2)类问题。
{"title":"Holographic Algorithms with Matchgates Capture Precisely Tractable Planar_#CSP","authors":"Jin-Yi Cai, P. Lu, Mingji Xia","doi":"10.1137/16M1073984","DOIUrl":"https://doi.org/10.1137/16M1073984","url":null,"abstract":"Valiant introduced match gate computation and holographic algorithms. A number of seemingly exponential time problems can be solved by this novel algorithmic paradigm in polynomial time. We show that, in a very strong sense, match gate computations and holographic algorithms based on them provide a universal methodology to a broad class of counting problems studied in statistical physics community for decades. They capture precisely those problems which are #P-hard on general graphs but computable in polynomial time on planar graphs. More precisely, we prove complexity dichotomy theorems in the framework of counting CSP problems. The local constraint functions take Boolean inputs, and can be arbitrary real-valued symmetric functions. We prove that, every problem in this class belongs to precisely three categories: (1) those which are tractable (i.e., polynomial time computable) on general graphs, or (2) those which are #P-hard on general graphs but ractable on planar graphs, or (3) those which are #P-hard even on planar graphs. The classification criteria are explicit. Moreover, problems in category (2) are tractable on planar graphs precisely by holographic algorithms with matchgates.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115005552","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 notion of {em a universally utility-maximizing privacy mechanism} was recently introduced by Ghosh, Rough garden, and Sundararajan~[STOC 2009]. These are mechanisms that guarantee optimal utility to a large class of information consumers, {em simultaneously}, while preserving {em Differential Privacy} [Dwork, McSherry, Nissim, and Smith, TCC 2006]. Ghosh, Rough garden and Sundararajan have demonstrated, quite surprisingly, a case where such a universally-optimal differentially-private mechanisms exists, when the information consumers are Bayesian. This result was recently extended by Gupte and Sundararajan~[PODS 2010] to risk-averse consumers. Both positive results deal with mechanisms (approximately) computing a {em single count query} (i.e., the number of individuals satisfying a specific property in a given population), and the starting point of our work is a trial at extending these results to similar settings, such as sum queries with non-binary individual values, histograms, and two (or more) count queries. We show, however, that universally-optimal mechanisms do not exist for all these queries, both for Bayesian and risk-averse consumers. For the Bayesian case, we go further, and give a characterization of those functions that admit universally-optimal mechanisms, showing that a universally-optimal mechanism exists, essentially, only for a (single) count query. At the heart of our proof is a representation of a query function $f$ by its {em privacy constraint graph} $G_f$ whose edges correspond to values resulting by applying $f$ to neighboring databases.
{"title":"Impossibility of Differentially Private Universally Optimal Mechanisms","authors":"H. Brenner, Kobbi Nissim","doi":"10.1137/110846671","DOIUrl":"https://doi.org/10.1137/110846671","url":null,"abstract":"The notion of {em a universally utility-maximizing privacy mechanism} was recently introduced by Ghosh, Rough garden, and Sundararajan~[STOC 2009]. These are mechanisms that guarantee optimal utility to a large class of information consumers, {em simultaneously}, while preserving {em Differential Privacy} [Dwork, McSherry, Nissim, and Smith, TCC 2006]. Ghosh, Rough garden and Sundararajan have demonstrated, quite surprisingly, a case where such a universally-optimal differentially-private mechanisms exists, when the information consumers are Bayesian. This result was recently extended by Gupte and Sundararajan~[PODS 2010] to risk-averse consumers. Both positive results deal with mechanisms (approximately) computing a {em single count query} (i.e., the number of individuals satisfying a specific property in a given population), and the starting point of our work is a trial at extending these results to similar settings, such as sum queries with non-binary individual values, histograms, and two (or more) count queries. We show, however, that universally-optimal mechanisms do not exist for all these queries, both for Bayesian and risk-averse consumers. For the Bayesian case, we go further, and give a characterization of those functions that admit universally-optimal mechanisms, showing that a universally-optimal mechanism exists, essentially, only for a (single) count query. At the heart of our proof is a representation of a query function $f$ by its {em privacy constraint graph} $G_f$ whose edges correspond to values resulting by applying $f$ to neighboring databases.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"41 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133095835","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 number of geometric permutations of an arbitrary collection of $n$ pair wise disjoint convex sets in $mathbb{R}^d$, for $dgeq 3$, is $O(n^{2d-3}log n)$, improving Wenger's 20 years old bound of $O(n^{2d-2})$.
{"title":"Improved Bounds for Geometric Permutations","authors":"Natan Rubin, Haim Kaplan, M. Sharir","doi":"10.1137/110835918","DOIUrl":"https://doi.org/10.1137/110835918","url":null,"abstract":"We show that the number of geometric permutations of an arbitrary collection of $n$ pair wise disjoint convex sets in $mathbb{R}^d$, for $dgeq 3$, is $O(n^{2d-3}log n)$, improving Wenger's 20 years old bound of $O(n^{2d-2})$.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115701941","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 notion of vertex sparsification (in particular cut-sparsification) is introduced in (Moitra, 2009), where it was shown that for any graph $G = (V, E)$ and any subset of $k$ terminals $K subset V$, there is a polynomial time algorithm to construct a graph $H = (K, E_H)$ emph{on just the terminal set} so that simultaneously for all cuts $(A, K-A)$, the value of the minimum cut in $G$ separating $A$ from $K -A$ is approximately the same as the value of the corresponding cut in $H$. Then approximation algorithms can be run directly on $H$ as a proxy for running on $G$. We give the first super-constant lower bounds for how well a cut-sparsifier $H$ can simultaneously approximate all minimum cuts in $G$. %In fact, we prove that in general we cannot hope for approximation factors better than We prove a lower bound of $Omega(log^{1/4} k)$ – this is polynomially-related to the known upper bound of $O(log k/log log k)$. Independently, a similar lower bound is given in (Makarychev, Makarychev, 2010). This is an exponential improvement on the $Omega(log log k)$ bound given in (Leighton, Moitra, 2010) which in fact was for a stronger vertex sparsification guarantee, and did not apply to cut sparsifiers. Despite this negative result, we show that for many natural optimization problems, we do not need to incur a multiplicative penalty for our reduction. Roughly, we show that any rounding algorithm which also works for the $0$-extension relaxation can be used to construct good vertex-sparsifiers for which the optimization problem is easy. Using this, we obtain optimal $O(log k)$-competitive Steiner oblivious routing schemes, which generalize the results in (Raecke, 2008). We also demonstrate that for a wide range of graph packing problems (which includes maximum concurrent flow, maximum multiflow and multicast routing, among others, as a special case), the integrality gap of the linear program is always at most $O(log k)$ times the integrality gap restricted to trees. Lastly, we use our ideas to give an efficient construction for vertex-sparsifiers that match the current best existential results – this was previously open. Our algorithm makes novel use of Earth-mover constraints.
在(Moitra, 2009)中引入了顶点稀疏化(特别是切点稀疏化)的概念，其中证明了对于任何图$G = (V, E)$和$k$终端$K subset V$的任何子集，存在一个多项式时间算法来在emph{终端集上}构建图$H = (K, E_H)$，以便同时对于所有切点$(A, K-A)$，$G$中分离$A$和$K -A$的最小切割量的值与$H$中相应的切割量的值大致相同。然后，近似算法可以直接在$H$上运行，作为在$G$上运行的代理。我们给出了cut- sparfier $H$同时近似$G$中所有最小cut的第一个超常数下界。 %In fact, we prove that in general we cannot hope for approximation factors better than We prove a lower bound of $Omega(log^{1/4} k)$ – this is polynomially-related to the known upper bound of $O(log k/log log k)$. Independently, a similar lower bound is given in (Makarychev, Makarychev, 2010). This is an exponential improvement on the $Omega(log log k)$ bound given in (Leighton, Moitra, 2010) which in fact was for a stronger vertex sparsification guarantee, and did not apply to cut sparsifiers. Despite this negative result, we show that for many natural optimization problems, we do not need to incur a multiplicative penalty for our reduction. Roughly, we show that any rounding algorithm which also works for the $0$-extension relaxation can be used to construct good vertex-sparsifiers for which the optimization problem is easy. Using this, we obtain optimal $O(log k)$-competitive Steiner oblivious routing schemes, which generalize the results in (Raecke, 2008). We also demonstrate that for a wide range of graph packing problems (which includes maximum concurrent flow, maximum multiflow and multicast routing, among others, as a special case), the integrality gap of the linear program is always at most $O(log k)$ times the integrality gap restricted to trees. Lastly, we use our ideas to give an efficient construction for vertex-sparsifiers that match the current best existential results – this was previously open. Our algorithm makes novel use of Earth-mover constraints.
{"title":"Vertex Sparsifiers and Abstract Rounding Algorithms","authors":"M. Charikar, F. Leighton, Shi Li, Ankur Moitra","doi":"10.1109/FOCS.2010.32","DOIUrl":"https://doi.org/10.1109/FOCS.2010.32","url":null,"abstract":"The notion of vertex sparsification (in particular cut-sparsification) is introduced in (Moitra, 2009), where it was shown that for any graph $G = (V, E)$ and any subset of $k$ terminals $K subset V$, there is a polynomial time algorithm to construct a graph $H = (K, E_H)$ emph{on just the terminal set} so that simultaneously for all cuts $(A, K-A)$, the value of the minimum cut in $G$ separating $A$ from $K -A$ is approximately the same as the value of the corresponding cut in $H$. Then approximation algorithms can be run directly on $H$ as a proxy for running on $G$. We give the first super-constant lower bounds for how well a cut-sparsifier $H$ can simultaneously approximate all minimum cuts in $G$. %In fact, we prove that in general we cannot hope for approximation factors better than We prove a lower bound of $Omega(log^{1/4} k)$ – this is polynomially-related to the known upper bound of $O(log k/log log k)$. Independently, a similar lower bound is given in (Makarychev, Makarychev, 2010). This is an exponential improvement on the $Omega(log log k)$ bound given in (Leighton, Moitra, 2010) which in fact was for a stronger vertex sparsification guarantee, and did not apply to cut sparsifiers. Despite this negative result, we show that for many natural optimization problems, we do not need to incur a multiplicative penalty for our reduction. Roughly, we show that any rounding algorithm which also works for the $0$-extension relaxation can be used to construct good vertex-sparsifiers for which the optimization problem is easy. Using this, we obtain optimal $O(log k)$-competitive Steiner oblivious routing schemes, which generalize the results in (Raecke, 2008). We also demonstrate that for a wide range of graph packing problems (which includes maximum concurrent flow, maximum multiflow and multicast routing, among others, as a special case), the integrality gap of the linear program is always at most $O(log k)$ times the integrality gap restricted to trees. Lastly, we use our ideas to give an efficient construction for vertex-sparsifiers that match the current best existential results – this was previously open. Our algorithm makes novel use of Earth-mover constraints.","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133555556","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 hardcore model is a model of lattice gas systems which has received much attention in statistical physics, probability theory and theoretical computer science. It is the probability distribution over independent sets $I$ of a graph weighted proportionally to $lambda^{|I|}$ with fugacity parameter $lambda$. We prove that at the uniqueness threshold of the hardcore model on the $d$-regular tree, approximating the partition function becomes computationally hard on graphs of maximum degree $d$. Specifically, we show that unless NP$=$RP there is no polynomial time approximation scheme for the partition function (the sum of such weighted independent sets) on graphs of maximum degree $d$ for fugacity $lambda_c(d) 0$. Weitz produced an FPTAS for approximating the partition function when $0
{"title":"Computational Transition at the Uniqueness Threshold","authors":"A. Sly","doi":"10.1109/FOCS.2010.34","DOIUrl":"https://doi.org/10.1109/FOCS.2010.34","url":null,"abstract":"The hardcore model is a model of lattice gas systems which has received much attention in statistical physics, probability theory and theoretical computer science. It is the probability distribution over independent sets $I$ of a graph weighted proportionally to $lambda^{|I|}$ with fugacity parameter $lambda$. We prove that at the uniqueness threshold of the hardcore model on the $d$-regular tree, approximating the partition function becomes computationally hard on graphs of maximum degree $d$. Specifically, we show that unless NP$=$RP there is no polynomial time approximation scheme for the partition function (the sum of such weighted independent sets) on graphs of maximum degree $d$ for fugacity $lambda_c(d) 0$. Weitz produced an FPTAS for approximating the partition function when $0","PeriodicalId":228365,"journal":{"name":"2010 IEEE 51st Annual Symposium on Foundations of Computer Science","volume":"490 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2010-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128089944","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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：