Abstract
Counting perfect matchings is an interesting and challenging combinatorial task. It has important applications in statistical physics. As the general problem is #P complete, it is usually tackled by randomized heuristics and approximation schemes. The trivial running times for exact algorithms are O *((n − 1)!!) = O *(n!!) = O *((n/2)! 2n/2) for general graphs and O *((n/2)!) for bipartite graphs. Ryser’s old algorithm uses the inclusion exclusion principle to handle the bipartite case in time O *(2n/2). It is still the fastest known algorithm handling arbitrary bipartite graphs.
For graphs with n vertices and m edges, we present a very simple argument for an algorithm running in time O *(1.4656m − n). For graphs of average degree 3 this is O *(1.2106n), improving on the previously fastest algorithm of Björklund and Husfeldt. We also present an algorithm running in time O *(1.4205m − n) or O *(1.1918n) for average degree 3 graphs. The purpose of these simple algorithms is to exhibit the power of the m − n measure.
Here, we don’t investigate the further improvements possible for larger average degrees by applying the measure-and-conquer method.
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References
Björklund, A., Husfeldt, T.: Exact algorithms for exact satisfiability and number of perfect matchings. Algorithmica 52(2), 226–249 (2008)
Björklund, A.: Counting perfect matchings as fast as Ryser. In: Proceedings of the Twenty-Third Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2012, pp. 914–921. SIAM (2012)
Dahllöf, V., Jonsson, P., Beigel, R.: Algorithms for four variants of the exact satisfiability problem. Theoretical Computer Science 320(2-3), 373–394 (2004)
Fürer, M.: A Faster Algorithm for Finding Maximum Independent Sets in Sparse Graphs. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 491–501. Springer, Heidelberg (2006)
Kullmann, O.: New methods for 3-SAT decision and worst-case analysis. Theoretical Computer Science 223, 1–72 (1999)
Ryser, H.J.: Combinatorial mathematics. Carus Math. Monographs, No. 14. Math. Assoc. of America, Washington, DC (1963)
Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8, 189–201 (1979)
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Fürer, M. (2012). Counting Perfect Matchings in Graphs of Degree 3. In: Kranakis, E., Krizanc, D., Luccio, F. (eds) Fun with Algorithms. FUN 2012. Lecture Notes in Computer Science, vol 7288. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-30347-0_20
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DOI: https://doi.org/10.1007/978-3-642-30347-0_20
Publisher Name: Springer, Berlin, Heidelberg
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