A combinatorial algorithm for Pfaffians

The Pfaffian of an oriented graph is closely linked to Perfect Matching. It is also naturally related to the determinant of an appropriately defined matrix. This relation between Pfaffian and determinant is usually exploited to give a fast algorithm for computing Pfaffians. We present the first completely combinatorial algorithm for computing the Pfaffian in polynomial time. Our algorithm works over arbitrary commutative rings. Over integers, we show that it can be computed in the complexity class GapL; this result was not known before. Our proof techniques generalize the recent combinatorial characterization of determinant [MV97] in novel ways. As a corollary, we show that under reasonable encodings of a planar graph, Kasteleyn's algorithm [Kas67] for counting the number of perfect matchings in a planar graph is also in GapL. The combinatorial characterization of Pfaffian also makes it possible to directly establish several algorithmic and complexity theoretic results on Perfect Matching which ..