Random generation of combinatorial structures from a uniform
Theoretical Computer Science
How hard is it to marry at random? (On the approximation of the permanent)
STOC '86 Proceedings of the eighteenth annual ACM symposium on Theory of computing
Conductance and the rapid mixing property for Markov chains: the approximation of permanent resolved
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
Approximate counting, uniform generation and rapidly mixing Markov chains
Information and Computation
A random polynomial time algorithm for approximating the volume of convex bodies
STOC '89 Proceedings of the twenty-first annual ACM symposium on Theory of computing
Efficient stopping rules for Markov chains
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
Graph orientations with no sink and an approximation for a hard case of #SAT
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Approximately counting Hamilton cycles in dense graphs
SODA '94 Proceedings of the fifth annual ACM-SIAM symposium on Discrete algorithms
A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries
Journal of the ACM (JACM)
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We give efficient randomized schemes to sample and approximately count Eulerian orientations of any Eulerian graph. Eulerian orientations are natural flow-like structures, and Welsh has pointed out that computing their number (i)corresponds to evaluating the Tutte polynomial at the point (0, –2) [8,19] and (ii) is equivalent to evaluating “ice-type partition functions” in statistical physics [20].Our algorithms are based on a reduction to sampling and approximately counting perfect matchings for a class of graphs for which the methods of Broder [3, 10] and others [4, 6] apply. A crucial step of the reduction is the “Monotonicity Lemma” (Lemma 3.3) which is of independent combinatorial interest. Roughly speaking, the Monotonicity Lemma establishes the intuitive fact that “increasing the number of constraints applied on a flow problem can only decrease the number of solutions”. In turn, the proof of the lemma involves a new decomposition technique which decouples problematically overlapping structures (a recurrent obstacle in handling large combinatorial populations) and allows detailed enumeration arguments. As a byproduct, (i) we exhibit a class of graphs for which perfect and near-perfect matchings are polynomially related, and hence the permanent can be approximated, for reasons other than “short augmenting paths” (previously the only known approach); and (ii) we obtain a further direct sampling scheme for Eulerian orientations which is faster than the one suggested by the reduction to perfect matchings.Finally, with respect to our approximate counting algorithm, we give the complementary hardness result, namely, that counting exactly Eulerian orientations is #P-complete, and provide some connections with Eulerian tours.