On the number of Eulerian orientations of a graph

Abstract

An “Eulerian orientation” of an undirected Eulerian graph is an orientation of the edges of the graph such that for every vertex the in-degree is equal to the out-degree. Eulerian orientations are natural flow-like structures, and Welsh has pointed out that computing their number corresponds to evaluating the Tutte polynomial at the point (0, −2) [JVW], [Wl], and is further equivalent to evaluating “ice-type partition functions” in statistical physics [W2]. In this paper we resolve the complexity of counting the number of Eulerian orientations of an arbitrary Eulerian graph. We give an efficient randomized approximation algorithm for counting Eulerian orientations of any Eulerian graph. Our algorithm is based on a reduction to counting perfect matchings for a class of graphs for which the methods of Broder [B], Jerrum and Sinclair [JS1], and others [DL] [DS] apply. A crucial step of the reduction is the “Monotonicity Lemma” (Lemma 3.1) 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 cannot increase the number of solutions.” 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 by-product, 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). We also give the complementary hardness result, namely, that counting exactly Eulerian orientations is #P-complete. Finally, we provide some connections with counting Euler tours.