Efficient algorithms for minimum-cost flow problems with piecewise-linear convex costs

We present two efficient algorithms for the minimum-cost flow problem in which arc costs are piecewise-linear and convex. Our algorithms are based on novel algorithms of Orlin, which were developed for the case of linear arc costs. Our first algorithm uses the Edmonds-Karp scaling technique. Its complexity is O(M log U(m+n log M)) for a network with n vertices, m arcs, M linear cost segments, and an upper bound U on the supplies and the capacities. The second algorithm is a strongly polynomial version of the first, and it uses Tardos's idea of contraction. Its complexity is O(M log M(m+n log M)). Both algorithms improve by a factor of at least M/m the complexity of directly applying existing algorithms to a transformed network in which arc costs are linear.