A Network Simplex Method for the Budget-Constrained Minimum Cost Flow Problem

TL;DR

This paper introduces a specialized network simplex algorithm for the budget-constrained minimum cost flow problem, extending traditional methods with novel potentials and costs, and demonstrates its effectiveness through theoretical analysis and computational experiments.

Contribution

It develops a fully combinatorial network simplex algorithm tailored for the budget-constrained minimum cost flow problem, incorporating new potentials and reduced costs.

Findings

Proves optimality criteria for the new algorithm.

Establishes a pseudo-polynomial running time for the procedure.

Computational results show competitive performance with Gurobi.

Abstract

We present a specialized network simplex algorithm for 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. We present a fully combinatorial description of the algorithm that is based on a novel incorporation of two kinds of integral node potentials and three kinds of reduced costs. We prove optimality criteria and combine two methods that are commonly used to avoid cycling in traditional network simplex algorithms into new techniques that are applicable to our problem. With these techniques and our definition of the reduced costs, we are able to prove a pseudo-polynomial running time of the overall procedure, which can be further improved by incorporating Dantzig's pivoting rule. Moreover, we present computational results that compare our procedure with Gurobi.