Scaling algorithms for network problems

A network is a graph with numeric parameters such as edge lengths, capacities, costs, etc. We present efficient algorithms for network problems that work by scaling the numeric parameters. Scaling takes advantage of efficient nonnumeric algorithms such as the Hopcroft-Karp matching algorithm. Let n, m and N denote the number of vertices, number of edges, and largest numeric parameter of the network, respectively; assume all numeric parameters are integers. A scaling algorithm for maximum weight matching on a bipartite graph runs in O(n3/4 m log N) time. This can improve the traditional Hungarian method which runs in O(n m log n) time. This result gives similar improvements for the following problems: single-source shortest paths for arbitrary edge lengths (Bellman's algorithm); maximum weight degree-constrained subgraph; minimum cost flow in a 0-1 network (Edmonds and Karp). Scaling also gives simple algorithms that match the best time bounds (when log N = O(log n)) for shortest paths on a directed graph with nonnegative lengths (Dijkstra's algorithm) and maximum value network flow (Sleator and Tarjan).