Budget-Constrained Minimum Cost Flows: The Continuous Case

Abstract

In this chapter, we study an extension of the well-known minimum cost flow problem in which a second kind of costs (called usage fees) is associated with each edge. The goal is to minimize the first kind of costs as in traditional minimum cost flows while the total usage fee of a flow must additionally fulfill a budget constraint. In the first part of this chapter, we present a specialized network simplex algorithm for the problem. In particular, we provide optimality criteria as well as measurements to avoid cycling. Moreover, we prove a pseudo-polynomial running time of the algorithm using Dantzig’s pivoting rule (cf. (Ahuja et al., 1988)). In the second part of the chapter, we present an interpretation of the problem as a bicriteria minimum cost flow problem, which allows us to obtain a weakly polynomial-time combinatorial algorithm that computes only ${\mathcal{O}}\left( {\log \,\text{M}} \right)$ traditional minimum cost flows, where M is the largest number that occurs in the problem instance. Moreover, we present a strongly polynomial-time algorithm that computes $\widetilde{{\mathcal{O}}}\left( {\text{nm}} \right)$ traditional minimum cost flows and derive three fully polynomial-time approximation schemes for the problem on general and on acyclic graphs.

This chapter is based on joint work with Sven O. Krumke and Clemens Thielen (Holzhauser et al., 2016a, 2015a, 2016b).