Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has proven profitable to decompose such objects into a structured component, a pseudo-random component, and a small component (i.e. an error term): in many cases it is the structured component which then dominates. We illustrate this philosophy in a number of model cases.
{"title":"Structure and Randomness in Combinatorics","authors":"T. Tao","doi":"10.1109/FOCS.2007.68","DOIUrl":"https://doi.org/10.1109/FOCS.2007.68","url":null,"abstract":"Combinatorics, like computer science, often has to deal with large objects of unspecified (or unusable) structure. One powerful way to deal with such an arbitrary object is to decompose it into more usable components. In particular, it has proven profitable to decompose such objects into a structured component, a pseudo-random component, and a small component (i.e. an error term): in many cases it is the structured component which then dominates. We illustrate this philosophy in a number of model cases.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-07-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126596425","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}
Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonAbelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.
{"title":"Quantum Algorithms for Hidden Nonlinear Structures","authors":"Andrew M. Childs, L. Schulman, U. Vazirani","doi":"10.1109/FOCS.2007.18","DOIUrl":"https://doi.org/10.1109/FOCS.2007.18","url":null,"abstract":"Attempts to find new quantum algorithms that outperform classical computation have focused primarily on the nonAbelian hidden subgroup problem, which generalizes the central problem solved by Shor's factoring algorithm. We suggest an alternative generalization, namely to problems of finding hidden nonlinear structures over finite fields. We give examples of two such problems that can be solved efficiently by a quantum computer, but not by a classical computer. We also give some positive results on the quantum query complexity of finding hidden nonlinear structures.","PeriodicalId":197431,"journal":{"name":"48th Annual IEEE Symposium on Foundations of Computer Science (FOCS'07)","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2007-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123199270","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 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.
{"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":null,"pages":null},"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}
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).
{"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":null,"pages":null},"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}
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