Strongly polynomial algorithm for a class of minimum-cost flow problems with separable convex objectives

• Authors: László A. Végh
• Venue: STOC '12 Proceedings of the forty-fourth annual ACM symposium on Theory of computing
• Year: 2012

A well-studied nonlinear extension of the minimum-cost flow problem is to minimize the objective ∑ij∈E Cij(fij) over feasible flows f, where on every arc ij of the network, Cij is a convex function. We give a strongly polynomial algorithm for finding an exact optimal solution for a broad class of such problems. The key characteristic of this class is that an optimal solution can be computed exactly provided its support. This includes separable convex quadratic objectives and also certain market equilibria problems: Fisher's market with linear and with spending constraint utilities. We thereby give the first strongly polynomial algorithms for separable quadratic minimum-cost flows and for Fisher's market with spending constraint utilities, settling open questions posed e.g. in [15] and in [35], respectively. The running time is O(m4 log m) for quadratic costs, O(n4+n2(m+n log n) log n) for Fisher's markets with linear utilities and O(mn3 +m2(m+n log n) log m) for spending constraint utilities.