The Complexity of Computing the Tutte Polynomial on Transversal Matroids

The complexity of computing the Tutte polynomial T(~/c,x,y) is determined for transversal matroid ,4s and algebraic numbers x and y. It is shown that for fixed x and y the problem of computing T(~,x,y) for JA a transversal matroid is ~pP-complete unless the numbers x and y satisfy (x 1)(y 1) = 1, in which case it is polynomial-time computable. In particular, the problem of counting bases in a transversal matroid, and of counting various types of "matchable" sets of nodes in a bipartite graph, is #P-complete.