A Decomposition Theorem for Maximum Weight Bipartite Matchings
  Let G be a bipartite graph with positive integer weights on the edges and without isolated nodes. Let n, N and W be the node count, the largest edge weight and the total weight of G. Let k(x,y) be log(x)/log(x^2/y). We present a new decomposition theorem for maximum weight bipartite matchings and use it to design an O(sqrt(n)W/k(n,W/N))-time algorithm for computing a maximum weight matching of G. This algorithm bridges a long-standing gap between the best known time complexity of computing a maximum weight matching and that of computing a maximum cardinality matching. Given G and a maximum weight matching of G, we can further compute the weight of a maximum weight matching of G-{u} for all nodes u in O(W) time. 
Ming-Yang Kao, Tak-Wah Lam, Wing-Kin Sung, Hing-Fung Ting