The complexity of counting graph homomorphisms (extended abstract)
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
Counting H-colorings of partial k-trees
Theoretical Computer Science
Counting H-Colorings of Partial k-Trees
COCOON '01 Proceedings of the 7th Annual International Conference on Computing and Combinatorics
ACM Transactions on Algorithms (TALG)
Towards a dichotomy theorem for the counting constraint satisfaction problem
Information and Computation
Some observations on holographic algorithms
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Sampling colourings of the triangular lattice
Random Structures & Algorithms
Efficient negative selection algorithms by sampling and approximate counting
PPSN'12 Proceedings of the 12th international conference on Parallel Problem Solving from Nature - Volume Part I
The complexity of the counting constraint satisfaction problem
Journal of the ACM (JACM)
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We consider approximate counting of colorings of an n-vertex graph using rapidly mixing Markov chains. It has been shown by Jerrum and by Salas and Sokal that a simple random walk on graph colorings would mix rapidly, provided the number of colors k exceeded the maximum degree $\Delta$ of the graph by a factor of at least 2. We prove that this is not a necessary condition for rapid mixing by considering the simplest case of 5-coloring graphs of maximum degree 3. Our proof involves a computer-assisted proof technique to establish rapid mixing of a new "heat bath" Markov chain on colorings using the method of path coupling. We outline an extension to 7-colorings of triangle-free 4-regular graphs. Since rapid mixing implies approximate counting in polynomial time, we show in contrast that exact counting is unlikely to be possible (in polynomial time). We give a general proof that the problem of exactly counting the number of proper k-colorings of graphs with maximum degree $\Delta$ is $\# P$-complete whenever $k\geq 3$ and $\Delta \geq 3$.