Fast Algorithms for Weighted Bipartite Matching

Let G = (V1 ∪ V2, E) be a bipartite graph on n nodes and m edges and let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$w : E \rightarrow {\mathbb R}_{+}$\end{document} be a weight function on the edges. We give several fast algorithms for computing a minimum weight (perfect) matching for a given complete bipartite graph (i.e. m = n2) by pruning the edge set. The algorithm will also output an upper bound on the achieved approximation factor. Under the assumption that the edge weights are uniformly distributed, we show that our algorithm will compute an optimal solution with high probability. From this we deduce an algorithm with fast expected running time that will always compute an optimal solution. For real edge weights we achieve a running time of O(n2logn) and for integer edge weights a running time of O(n2).