A Combinatorial Algorithm for Pfa

The Pfaaan of a graph is closely linked to Perfect Matching. It is also naturally related to the determinant of an appropriately deened matrix. This relation between Pfaaan and determinant is usually exploited to give a fast algorithm for computing Pfaaans. We present the rst completely combinatorial algorithm for computing the Pfaaan in polynomial time. In fact, 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 for counting the number of perfect match-ings in a planar graph is also in GapL.The combinatorial characterization of Pfaaan also makes it possible to directly establish several algorithmic and complexity theoretic results on Perfect Matching which otherwise use determinants in a roundabout way.