Approximation algorithms for counting the number of perfect matchings in bipartite graphs∗

The problem of devising an algorithm for counting the number of perfect matchings in bipartite graphs has a long history. Apparently the first algorithm (which works only in planar graphs) was presented in 1961 [8]. Then Valiant in 1979 showed that the problem is #P-complete, meaning that no polynomial-time algorithm exists for the problem, unless P = NP [11]. After that, people focused on developing fast approximation algorithms for the problem. However, even developing a polynomial-time approximation algorithm appeared to be very difficult, as the first fully polynomial-time approximation scheme (see Section 3 for definition) for the problem was given in 2001 [6], and it uses a rather complicated Markov chain. In Table 1, I have collected some of the important results. For each result, the class of graphs in which the algorithm works and the running time are mentioned. Some notations need to be defined to read the table properly: the graph in question is a bipartite graph with two parts of size n, we say the graph is γ-dense if the degree of each vertex is at least γn, and the O∗ notation hides the log n and −1 (where is the maximum acceptable error of the algorithm) factors. In this report I will present the algorithm of Jerrum and Sinclair for 1 2 -dense graphs, and then provide a summary of the exponential algorithm of Jerrum and U. Vazirani. Section 2 is an introduction to Markov chains for those who have no previous background. If you have a basic knowledge about Markov chains you can skip Section 2. In Section 3 the terminology is introduced. In Section 4, which is the longest section, the first algorithm ∗This is a report for the course CO 754: Approximation Algorithms, Winter 2010, University of Waterloo. †[email protected]