Efficient algorithms for maximum weight matchings in general graphs with small edge weights

Let G = (V, E) be a graph with positive integral edge weights. Our problem is to find a matching of maximum weight in G. We present a simple iterative algorithm for this problem that uses a maximum cardinality matching algorithm as a subroutine. Using the current fastest maximum cardinality matching algorithms, we solve the maximum weight matching problem in O(W √nm logn(n2/m)) time, or in O(Wnω) time with high probability, where n = |V|, m = |E|, W is the largest edge weight, and ω < 2.376 is the exponent of matrix multiplication. In relatively dense graphs, our algorithm performs better than all existing algorithms with W = o(log1.5 n). Our technique hinges on exploiting Edmonds' matching polytope and its dual.

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