Approximate counting, uniform generation and rapidly mixing Markov chains
Information and Computation
SIAM Journal on Computing
The Complexity of Counting in Sparse, Regular, and Planar Graphs
SIAM Journal on Computing
Markov Chain Algorithms for Planar Lattice Structures
SIAM Journal on Computing
Path coupling: A technique for proving rapid mixing in Markov chains
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries
Journal of the ACM (JACM)
Combinatorial Enumeration
Random sampling of 3-colorings in ℤ2
Random Structures & Algorithms - Isaac Newton Institute Programme “Computation, Combinatorics and Probability”: Part I
Path coupling without contraction
Journal of Discrete Algorithms
Accelerating Simulated Annealing for the Permanent and Combinatorial Counting Problems
SIAM Journal on Computing
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We consider the problem of sampling from the uniform distribution on the set of Eulerian orientations of subgraphs of the triangular lattice. Although Mihail and Winkler (1989) showed that this can be achieved in polynomial time for any graph, the algorithm studied here is more natural in the context of planar Eulerian graphs. We analyse the mixing time of a Markov chain on the Eulerian orientations of a planar graph which moves between orientations by reversing the edges of directed faces. Using path coupling and the comparison method we obtain a polynomial upper bound on the mixing time of this chain for any solid subgraph of the triangular lattice. By considering the conductance of the chain we show that there exist non-solid subgraphs (subgraphs with holes) for which the chain will always take an exponential amount of time to converge. Finally, we show that the problem of counting Eulerian orientations remains #P-complete when restricted to planar graphs (Mihail and Winkler had already established this for general graphs).