Convex generalized flows

Abstract

We study an extension of the well-known generalized maximum flow problem in which the outflow of an edge is a strictly increasing convex function of its inflow. In contrast to the traditional generalized maximum flow problem, in which the outflow of an edge depends linearly on its inflow and which is solvable in polynomial time, we show that, for convex outflow functions, the problem becomes strongly $\mathcal{NP}$-hard to solve even on bipartite acyclic graphs and weakly $\mathcal{NP}$-hard on series–parallel graphs. Both results hold even if the outflow functions are convex quadratic. Furthermore, we present (exponential-time) exact algorithms for computing optimal flows on acyclic and series–parallel graphs and optimal preflows on general graphs. We also show that a flow decomposition similar to the one for traditional generalized flows is possible and present a pseudo-polynomial-time algorithm for the case of integral flows on series–parallel graphs.