We present the first algorithm for generating random variates exactly uniformly from the set of perfect matchings of a bipartite graph with a polynomial expected running time over a nontrivial set of graphs. Previous Markov chain results obtain approximately uniform variates for arbitrary graphs in polynomial time, but their general running time is (n 10 (ln n) 2 ). Other algorithms (such as Kasteleyn's O(n 3 ) algorithm for planar graphs) concentrated on restricted versions of the problem. Our

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... algorithm employs acceptance/rejection together with a new upper limit on the permanent of a form similar to Bregman's theorem. For graphs with 2n nodes, where the degree of every node is γ n for a constant γ , the expected running time is O(n 1.5+.5/γ ). Under these conditions, Jerrum and Sinclair showed that a Markov chain of Broder can generate approximately uniform variates in (n 4.5+.5/γ ln n) time, making our algorithm significantly faster on this class of graphs. The problem of counting the number of perfect matchings in these types of graphs is P complete. In addition, we give a 1 + σ approximation algorithm for finding the permanent of 0-1 matrices with identical row and column sums that runs in O(n 1.5+.5/γ (1/σ 2 ) log(1/δ)), where the probability that the output is within 1 + σ of the permanent is at least 1 − δ.