A strongly polynomial algorithm for minimum cost submodular flow problems

The only known strongly polynomial algorithm for solving minimum cost submodular flow problems is due to Frank and Tardos and is based on the simultaneous approximation algorithm of Lenstra, Lenstra, and Lovász. The authors propose a purely combinatorial strongly polynomial algorithm. It consists in solving a sequence of at most m+n(n-1) minimum cost submodular flow problems with cost coefficients bounded by n², where n is the number of the vertices and m is the number of the arcs in the underlying graph. The current cost coefficients are calculated by means of tree projection and scaling.