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Generating a random sink-free orientation in quadratic time

Henry Cohn·Robin Pemantle·James Propp
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Abstract

A sink-free orientation of a finite undirected graph is a choice of orientation for each edge such that every vertex has out-degree at least 1. Bubley and Dyer (1997) use Markov Chain Monte Carlo to sample approximately from the uniform distribution on sink-free orientations in time O(m^3 log (1/epsilon)), where m is the number of edges and epsilon the degree of approximation. Huber (1998) uses coupling from the past to obtain an exact sample in time O(m^4). We present a simple randomized algorithm inspired by Wilson's cycle popping method which obtains an exact sample in mean time at most O(nm), where n is the number of vertices.

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Henry Cohn, Robin Pemantle, James Propp. 2002-03-29. Generating a random sink-free orientation in quadratic time. https://arxiv.org/abs/math/0103189

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