Counting the Number of Eulerian Orientations

Consider an undirected Eulerian graph, a graph in which each vertex has even degree. An Eulerian orientation of the graph is an orientation of its edges such that for each vertex v, the number of incoming edges of v equals to outgoing edges of v, i.e. din(v) = dout(v). Let P0 denote the set of all Eulerian orientations of graph G. In this paper, we are concerned with the questions of sampling uniformly and counting the set of P0 of an arbitrary graph G. The significance of counting the number of Eulerian orientation was raised in several different fields. In statistical physics, the crucial partition function “ZICE”, in the so-called “ice-type model” [5], is equal to the number of Eulerian orientations of some underlying Eulerian graph. It has been also observed [2, 4] that the counting problem for Eulerian orientations corresponds to evaluating the Tutte polynomial at the point (0,−2). In addition, If the Eulerian graph G represents the topology of a network, an Eulerian orientation is a unidirectional configuration of the network, which is a maximum flow without source and sink. Consequently, counting the number of Eulerian orientations is equivalent to counting the number of maximum flows around the network. Finding an Eulerian orientation of a graph G can be accomplished in polynomial time, but sampling and counting the set of all Eulerian orientations is intractable, unless unexpected collapses of complexity classes occur [1]. This behavior of a problem was first observed in the case of perfect matchings of bipartite graphs. For perfect matchings, construction is well known to be solvable in polynomial time, but exact enumeration is complete for the class #P [3].The latter hardness result directed efforts toward obtaining efficient randomized approximations. Now, sampling and approximate counting can be achieved via randomized schemes which run in time polynomial in the size of the input graph, the inverse of the desired approximation accuracy, and the ratio of the total number of near-perfect over perfect matchings |Mn−1|/|Mn| of the input graph. In this paper, we show that sampling and approximately counting Eulerian orientations reduce to sampling and approximately counting perfect matchings for a class of graphs where |Mn−1|/|Mn| = O(n). Thus, we obtain efficient solutions for Eulerian orientations. This reduction is a fully combinatorial argument without resorting to any theory of Markov chains.