A scaling algorithm for weighted matching on general graphs

This paper presents an algorithm for maximum matching on general graphs with integral edge weights, running in time O(n3/4m lg N), where n, m and N are the number of vertices, number of edges, and largest edge weight magnitude, respectively. The best previous bound is O(n(mlg lg lgd n + n lg n)) where d is the density of the graph. The algorithm finds augmenting paths in batches by scaling the weights. The algorithm extends to degree-constrained subgraphs and hence to shortest paths on undirected graphs, the Chinese postman problem and finding a maximum cut of a planar graph. It speeds up Christofides' travelling salesman approximation algorithm from O(n3) to O(n2.75 lg n). A list splitting problem that arises in Edmonds' matching algorithm is solved in O(mα(m,n)) time, where m is the number of operations on a universe of n elements; the list splitting algorithm does not use set merging. Applications are given to update problems for red-green matching, the cardinality Chinese postman problem and the maximum cardinality plane cut problem; also to the all-pairs shortest paths problem on undirected graphs with lengths plus or minus one.