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(n /m)) time, or in O(Wn) 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(log n). Our technique hinges on exploiting Edmonds’ matching polytope and its dual.