48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)最新文献_第6页

Structure and Randomness in Combinatorics 组合学中的结构与随机性

48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07) Pub Date : 2007-07-29 DOI: 10.1109/FOCS.2007.68

T. Tao

引用次数: 48

Quantum Algorithms for Hidden Nonlinear Structures 隐非线性结构的量子算法

48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07) Pub Date : 2007-05-21 DOI: 10.1109/FOCS.2007.18

Andrew M. Childs, L. Schulman, U. Vazirani

引用次数: 79

Adaptive Simulated Annealing: A Near-optimal Connection between Sampling and Counting 自适应模拟退火:采样和计数之间的近最优连接

48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07) Pub Date : 2006-12-10 DOI: 10.1145/1516512.1516520

Daniel Stefankovic, S. Vempala, Eric Vigoda

{"title":"Adaptive Simulated Annealing: A Near-optimal Connection between Sampling and Counting","authors":"Daniel Stefankovic, S. Vempala, Eric Vigoda","doi":"10.1145/1516512.1516520","DOIUrl":"https://doi.org/10.1145/1516512.1516520","url":null,"abstract":"We present a near-optimal reduction from approximately counting the cardinality of a discrete set to approximately sampling elements of the set. An important application of our work is to approximating the partition function Z of a discrete system, such as the Ising model, matchings or colorings of a graph. The standard approach to estimating the partition function Z(beta*) at some desired inverse temperature beta* is to define a sequence, which we call a cooling schedule, beta0 = 0 < beta1 < ldrldrldr < betal = beta* where Z(0) is trivial to compute and the ratios Z(betai+1)/Z(betai) are easy to estimate by sampling from the distribution corresponding to Z(betai). Previous approaches required a cooling schedule of length O*(ln A) where A = Z(0), thereby ensuring that each ratio Z(betai+1)/Z(betai) is bounded. We present a cooling schedule of length l = O*(radicln A). For well-studied problems such as estimating the partition function of the Ising model, or approximating the number of colorings or matchings of a graph, our cooling schedule is of length O* (radicln) and the total number of samples required is O*(n). This implies an overall savings of a factor of roughly n in the running time of the approximate counting algorithm compared to the previous best approach. A similar improvement in the length of the cooling schedule was recently obtained by Lovtisz and Vempala in the context of estimating the volume of convex bodies. While our reduction is inspired by theirs, the discrete analogue of their result turns out to be significantly more difficult. Whereas a fixed schedule suffices in their setting, we prove that in the discrete setting we need an adaptive schedule, i. e., the schedule depends on Z. More precisely, we prove any non-adaptive cooling schedule has length at least O*(ln A), and we present an algorithm to find an adaptive schedule of length O* (radicln A) and a nearly matching lower bound.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":"23 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129596045","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}

引用次数: 118

Paths Beyond Local Search: A Tight Bound for Randomized Fixed-Point Computation 超越局部搜索的路径:随机不动点计算的紧界

48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07) Pub Date : 1900-01-01 DOI: 10.1109/FOCS.2007.14

X. Chen, S. Teng

{"title":"Paths Beyond Local Search: A Tight Bound for Randomized Fixed-Point Computation","authors":"X. Chen, S. Teng","doi":"10.1109/FOCS.2007.14","DOIUrl":"https://doi.org/10.1109/FOCS.2007.14","url":null,"abstract":"In 1983, Akhus proved that randomization can speedup local search. For example, it reduces the query complexity of local search over grid [1 : n]d from ominus(nd-1) to 0(d1/2nd/2). It remains open whether randomisation helps fixed-point computation. Inspired by the recent advances on the complexity of equilibrium computation, we solve this open problem by giving an asymptotically tight bound of (Omega(n))d-1 on the randomized query complexity for computing a fixed point of a discrete Brouwer function over grid [1 : n]d. Our result can be extended to the black-box query model for Sperner's I&mma in any dimension. It also yields a tight bound for the computation of d-dimensional approximate Brouwer fixed points as defined by Scarf and by Hirsch, Papadimitriou, and Vavasis. Since the randomized query complexity of global optimization over [1 : n]d is ominus(nd), the randomized query model over [ 1 : n]d strictly separates these three important search problems: Global optimization is harder than fixed-point computation, and fixed-point computation is harder than local search. Our result indeed demonstrates that randomization does not help much in fixed-point computation in the black-box query model. Our randomized lower bound matches the deterministic complexity of this problem, which is ominus(nd-1).","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":"406 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132623276","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}

引用次数: 17