On the Complexity and Approximability of Budget-Constrained Minimum Cost Flows

TL;DR

This paper studies the complexity and approximation of a budget-constrained minimum cost flow problem, providing exact algorithms and approximation schemes for general and acyclic graphs.

Contribution

It introduces a weakly polynomial-time exact algorithm and multiple fully polynomial-time approximation schemes for the problem.

Findings

Exact solution in weakly polynomial time with $O( ext{log} M imes ext{MCF}(m,n,C,U))$ complexity.

Two FPTAS developed for general graphs.

Improved FPTAS for acyclic graphs.

Abstract

We investigate the complexity and approximability of the budget-constrained minimum cost flow problem, which is an extension of the traditional minimum cost flow problem by a second kind of costs associated with each edge, whose total value in a feasible flow is constrained by a given budget B. This problem can, e.g., be seen as the application of the {\epsilon}-constraint method to the bicriteria minimum cost flow problem. We show that we can solve the problem exactly in weakly polynomial time $O(\log M \cdot MCF(m,n,C,U))$, where C, U, and M are upper bounds on the largest absolute cost, largest capacity, and largest absolute value of any number occuring in the input, respectively, and MCF(m,n,C,U) denotes the complexity of finding a traditional minimum cost flow. Moreover, we present two fully polynomial-time approximation schemes for the problem on general graphs and one with an improved running-time for the problem on acyclic graphs.