Budget-Constrained Minimum Cost Flows: The Discrete Case

Abstract

We now consider two discrete variants of the budget-constrained minimum cost flow problem that was investigated in the previous chapter. We show that both variants may be interpreted as network improvement problems, which yields several fields of applications. As a first variant, we prove that the problem becomes both weakly ${\mathcal{N}\mathcal{P}}$-hard to solve and approximate if the usage fees are induced in integral units. For the case of series-parallel graphs, we derive a pseudo-polynomial-time exact algorithm. Moreover, we present an interesting interpretation of the problem on extension-parallel graphs as a knapsack problem and provide both an approximation algorithm and (fully) polynomialtime approximation schemes. Finally, as a second discrete variant, we investigate a binary case in which a fee is incurred for a positive flow on an edge, independently of the flow’s magnitude. For this case, the problem becomes strongly ${\mathcal{N}\mathcal{P}}$-hard to solve, but still solvable in pseudo-polynomial-time on series-parallel graphs and easy to approximate under several restrictions on extension-parallel graphs.

This chapter is based on joint work with Sven O. Krumke and Clemens Thielen (Holzhauser et al., 2016a).