Random generation of combinatorial structures from a uniform
Theoretical Computer Science
SIAM Journal on Computing
A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
A very simple algorithm for estimating the number of k-colorings of a low-degree graph
Random Structures & Algorithms
Random walks and an O*(n5) volume algorithm for convex bodies
Random Structures & Algorithms
A Chernoff Bound for Random Walks on Expander Graphs
SIAM Journal on Computing
A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries
Journal of the ACM (JACM)
Mixing in time and space for discrete spin systems
Mixing in time and space for discrete spin systems
Counting independent sets up to the tree threshold
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Simulated annealing in convex bodies and an O*(n4) volume algorithm
Journal of Computer and System Sciences - Special issue on FOCS 2003
Fast Algorithms for Logconcave Functions: Sampling, Rounding, Integration and Optimization
FOCS '06 Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science
Accelerating Simulated Annealing for the Permanent and Combinatorial Counting Problems
SIAM Journal on Computing
Improved inapproximability results for counting independent sets in the hard-core model
APPROX'11/RANDOM'11 Proceedings of the 14th international workshop and 15th international conference on Approximation, randomization, and combinatorial optimization: algorithms and techniques
Submodular maximization by simulated annealing
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
A flat histogram method for computing the density of states of combinatorial problems
IJCAI'11 Proceedings of the Twenty-Second international joint conference on Artificial Intelligence - Volume Volume Three
A Deterministic Polynomial-Time Approximation Scheme for Counting Knapsack Solutions
SIAM Journal on Computing
Counting subsets of contingency tables
Computational Statistics
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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 typical approach to estimating the partition function Z(β*) at some desired inverse temperature β* is to define a sequence, which we call a cooling schedule, β0 = 0 1 ℓ = β* where Z(0) is trivial to compute and the ratios Z(βi+1)/Z(βi) are easy to estimate by sampling from the distribution corresponding to Z(βi). Previous approaches required a cooling schedule of length O*(ln A) where A=Z(0), thereby ensuring that each ratio Z(βi+1)/Z(βi) is bounded. We present a cooling schedule of length ℓ =O*(&sqrt; lnA). 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*(&sqrt; n), which implies an overall savings of O*(n) in the running time of the approximate counting algorithm (since roughly ℓ samples are needed to estimate each ratio). A similar improvement in the length of the cooling schedule was recently obtained by Lovász 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, that is, the schedule depends on Z. More precisely, we prove any nonadaptive cooling schedule has length at least O*(ln A), and we present an algorithm to find an adaptive schedule of length O*(&sqrt; ln A).