Sampling Eulerian orientations of triangular lattice graphs

We consider the problem of sampling from the uniform distribution on the set of Eulerian orientations of subgraphs of the triangular lattice. Although it is known 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 subgraphs with holes for which the chain will always take an exponential amount of time to converge. Finally, as an additional justification for studying a Markov chain on the set of Eulerian orientations of planar graphs, we show that the problem of counting Eulerian orientations remains #P-complete when restricted to planar graphs. A preliminary version of this work appeared as an extended abstract in the 2nd Algorithms and Complexity in Durham workshop.