Scaling Algorithms for Network Problems

This paper gives algorithms for network problems that work by scaling the numeric parameters. Assume all parameters are integers. Let n, m, and N denote the number of vertices, number of edges, and largest parameter of the network, respectively. A scaling algorithm for maximum weight matching on a bipartite graph runs in O(n34m log N) time. For appropriate N this improves the traditional Hungarian method, whose most efficient implementation is O(n(m +n log n)). The speedup results from finding augmenting paths in batches. The matching algorithm 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 on a 0–1 network. Scaling gives a simple maximum value flow algorithm that matches the best known bound (Sleator and Tarjan's algorithm) when log N=O(log n). Scaling also gives a good algorithm for shortest paths on a directed graph with nonnegative edge lengths (Dijkstra's algorithm).