Convex Generalized Flows

Abstract

In this chapter, we give insights into the structural properties and the complexity of an extension of the 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, which is solvable in polynomial time as shown in Section 2.4, we show that the problem becomes NP-hard to solve and approximate in this novel setting. Nevertheless, we show that a flow decomposition similar to the one for traditional generalized flows is possible and present (exponential-time) exact algorithms for computing optimal flows on acyclic, series-parallel, and extension-parallel graphs as well as optimal preflows on general graphs. We also identify a polynomially solvable special case and show that the problem is solvable in pseudo-polynomial time when restricting to integral flows on series-parallel graphs.

This chapter is based on joint work with Sven O. Krumke and Clemens Thielen (Holzhauser et al., 2015b).