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

Abstract

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(m 4 log m) for quadratic costs, O(n 4 +n 2 (m+n log n) log n) for Fisher's markets with linear utilities and O(mn 3 +m 2 (m+ n log n) log m) for spending constraint utilities.

A strongly polynomial algorithm for linear exchange markets

“…Our approach was specific to the market equilibrium problem. The method of identifying revealed arc sets originates from [39]. This result was applicable not only for the linear Fisher market model, but more generally, for minimum-cost flow problems with separable convex objectives satisfying certain assumptions.…”

“…In the algorithm, we maintain a set F ⊆ F * called revealed edges, and the main progress is adding a new edge in strongly polynomial time. At a high level, this approach resembles that of [39], which extends Orlin's approach for minimum-cost circulations [29].…”

“…The linear Fisher market equilibrium can be captured by two different convex programs, one by Eisenberg and Gale [15], and one by Shmyrev [32]. These are special cases of natural convex extensions of classical network flow models [38,39]. In particular, the second model is a network flow problem with a separable convex cost function; [39] provides a strongly polynomial algorithm for the linear Fisher market using this general perspective.…”

“…Let us now discuss the Price Boost subroutine. The analogous subproblem for Fisher markets in [39] reduces to a simple variant of the Floyd-Warshall algorithm. For exchange markets, we show that optimizing φ(p, f ) can be captured by a linear program.…”

Approximating the Nash Social Welfare with Budget-Additive Valuations

“…These equilibria often can be computed by solving convex programs due to Eisenberg and Gale [26] or Shmyrev [47]. For additive valuations, there are combinatorial [22] and even strongly polynomialtime algorithms [44,51] for computing such an equilibrium.…”