TY - GEN

T1 - Counting perfect matchings in graphs of degree 3

AU - Furer, Martin

PY - 2012/6/13

Y1 - 2012/6/13

N2 - 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)! 2 n/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*(2 n/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 *(1.4656 m-n ). For graphs of average degree 3 this is O*(1.2106 n ), improving on the previously fastest algorithm of Björklund and Husfeldt. We also present an algorithm running in time O*(1.4205 m-n ) or O*(1.1918 n ) 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.

AB - 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)! 2 n/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*(2 n/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 *(1.4656 m-n ). For graphs of average degree 3 this is O*(1.2106 n ), improving on the previously fastest algorithm of Björklund and Husfeldt. We also present an algorithm running in time O*(1.4205 m-n ) or O*(1.1918 n ) 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.

UR - http://www.scopus.com/inward/record.url?scp=84861969856&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84861969856&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-30347-0_20

DO - 10.1007/978-3-642-30347-0_20

M3 - Conference contribution

AN - SCOPUS:84861969856

SN - 9783642303463

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 189

EP - 197

BT - Fun with Algorithms - 6th International Conference, FUN 2012, Proceedings

T2 - 6th International Conference on Fun with Algorithms, FUN 2012

Y2 - 4 June 2012 through 6 June 2012

ER -