Random generation of combinatorial structures from a uniform
Theoretical Computer Science
A very simple algorithm for estimating the number of k-colorings of a low-degree graph
Random Structures & Algorithms
Gibbs measures and dismantlable graphs
Journal of Combinatorial Theory Series B
The complexity of counting graph homomorphisms
Proceedings of the ninth international conference on on Random structures and algorithms
On Markov chains for randomly H-coloring a graph
Journal of Algorithms
The complexity of choosing an H-colouring (nearly) uniformly at random
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Rapidly Mixing Markov Chains for Dismantleable Constraint Graphs
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
Foundations of Cryptography: Volume 1
Foundations of Cryptography: Volume 1
Rapidly Mixing Markov Chains for Dismantleable Constraint Graphs
RANDOM '02 Proceedings of the 6th International Workshop on Randomization and Approximation Techniques
The complexity of partition functions
Theoretical Computer Science - Automata, languages and programming: Algorithms and complexity (ICALP-A 2004)
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For counting problems in #P which are "essentially self-reducible", it is known that sampling and approximate counting are equivalent. However, many problems of interest do not have such a structure and there is already some evidence that this equivalence does not hold for the whole of #P. An intriguing example is the class of H- colouring problems, which have recently been the subject of much study, and their natural generalisation to vertex- and edge-weighted versions. Particular cases of the counting-to-sampling reduction have been observed, but it has been an open question as to how far these reductions might extend to any H and a general graph G. Here we give the first completely general counting-to-sampling reduction. For every fixed H, we show that the problem of approximately determining the partition function of weighted H-colourings can be reduced to the problem of sampling these colourings from an approximately correct distribution. In particular, any rapidly-mixing Markov chain for sampling H-colourings can be turned into an FPRAS for counting H-colourings.