Estimating the permanent
نویسنده
چکیده
We refer the reader to Jerrum's book [1] for the analysis of a Markov chain for generating a random matching of an arbitrary graph. Here we'll look at how to extend the argument to sample perfect matchings in dense graphs and arbitrary bipartite graphs. Both of these results are due to Jerrum and Sinclair [2]. Dense graphs We'll need some definitions and notations first: Definition 7.1 A graph G = (V, E) is said to be dense if for every v ∈ V , degree(v) > n/2, where n = |V |. Definition 7.2 Let G be a graph and let M G be the set of all perfect matchings of G. Similarly, for two distinct vertices u, v, let N G (u, v) be the set of all perfect matchings of the graph G \ {u, v}. We refer to matchings in N G (u, v) as near-perfect matchings, or matchings with holes u and v. Whenever G is clearly understood from the context, we shall use simply M and N (u, v). Jerrum's book presents the analysis of a Markov chain for sampling matchings (not necessarily perfect). The Markov chain for sampling perfect matchings in dense graphs is only a slight modification of the original chain. The state space Ω is the set of perfect and near-perfect matchings, i.e., Ω = M ∪ ∪ u,v∈V N (u, v). The near-perfect matchings are needed to navigate between perfect matchings.
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