Multi-player minimum cost flow problems with nonconvex costs and integer flows

In this paper we consider a variant of the well known minimum cost flow problem in a directed network with nonconvex costs and integer flows. We formulate the problem in a multi-player setup, whereby we associate one player with each arc of the network. The goal of each player is to minimize its nonconvex cost that depends on the integer flow through the arc subject to the network flow constraints. In this multi-player setup, a Pareto optimal point is justified to be an efficient solution concept. We propose an algorithm to compute a Pareto optimal point. We show that, although the problem in its original form has coupled constraints binding every player, there exists an equivalent variable transformation that decouples the optimization problems for a number of players. Each of the decoupled players can solve its optimization problem in a decentralized manner. We use the solutions of those decoupled players to transform the optimization problems for the rest of the players using consensus constraints. Then we present algorithms based on algebraic geometry to find a Pareto optimal point.