and Tel-Aviv University

The minimum-cost flow problem is the following: given a network with n vertices and m edges, find a maximum flow of minimum cost. ~lany network problems are easily reducible to this problem. A. polynomial-time algorithm for the problem has been known for some time, but only recently a strongly polynomial algorithm was discovered. In this paper we design an O ( n2 ( m + n log n) log n) algorithm. The previous best algorithm had an O ( m 2 ( m+n log n) log n) time bound. Thus, we obtain an improvement of two orders of magnitude for dense graphs. Our algorithm is based on Fujishige's algorithm (which is based on Tardos ' algorithm. Fujishige's algorithm consists of up to m iterations, each consisting of O ( m log n) steps. Each step solves a single source shortest path problem with nonnegative edge lengths. \Ve modify this algorithm in order to make an improved analysis possible. The new algorithm may still consist of up to m iterations, and an iteration may still consist of up to O ( m log n) steps, but we can still show that the total number of steps is bounded by O ( n 2 log n). The improvement is due to a new technique that relates the time spent to the progress achieved. 1