A combinatorial algorithm for Pfaffians

Authors:
Meena Mahajan;P. R. Subramanya;V. Vinay
Affiliations:
nstitute of Mathematical Sciences, Chennai, India;Dept. of Computer Science & Automation, Indian Institute of Science, Bangalore, India;Dept. of Computer Science & Automation, Indian Institute of Science, Bangalore, India
Venue:
COCOON'99 Proceedings of the 5th annual international conference on Computing and combinatorics
Year:
1999

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Abstract

The Pfaffian of a 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. 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 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 otherwise use determinants in a roundabout way.