A Combinatorial Algorithm for Pfaffians
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.
KeywordsPlanar Graph Perfect Match Edge Weight Combinatorial Algorithm Simple Cycle
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- AlRe98.E. Allender and K. Reinhardt, Isolation, Matching, and Counting, in IEEE Conference on Computational Complexity, pp 92–100, 1998.Google Scholar
- Kas67.P. W. Kasteleyn, Graph Theory and Crystal Physics, in Graph Theory and Theoretical Physics, ed: F. Harary, Academic Press, pp 43–110, 1967.Google Scholar
- Lit74.C. H. C. Little, An extension of Kasteleyn's method of enumerating the 1-factors of planar graphs, in Combinatorial Mathematics, Proc. 2nd Australian Conference, ed: D. Holton, Lecture Notes in Mathematics, 403, pp 63–72, 1974.Google Scholar
- LovPlu86.L. Lovasz and M. Plummer, Matching Theory, Annals of Discrete Mathematics 29, North-Holland, 1986.Google Scholar
- MV97.M. Mahajan and V. Vinay, Determinants: Combinatorics, algorithms, and complexity, Chicago Journal of Theoretical Computer Science, vol. 5, 1997.Google Scholar
- MRST97.W. McCuaig, N. Robertson, P. D. Seymour and R. Thomas, Permanents, Pfaffian Orientations, and Even Directed Circuits, Proc. of IEEE Symposium on Theory of Computing, pp 402–405, 1997.Google Scholar
- Vaz89.V. V. Vazirani, NC algorithms for computing the number of perfect matchings in K 3,3 -free graphs and related problems, Information and Computation, vol. 80(2), pp 152-164, 1989.Google Scholar
- Vin91.V. Vinay, Counting auxiliary pushdown automata and semi-unbounded arithmetic circuits, Proc. 6th IEEE Structure in Complexity Theory Conference, pp 270–284, 1991.Google Scholar