A Combinatorial Algorithm for the Multi-commodity Flow Problem

TL;DR

This paper introduces a combinatorial algorithm for the multi-commodity flow problem using a penalty-based approach and equilibrium pseudo-flow concepts, providing a new perspective on feasibility and solution characterization.

Contribution

It proposes a novel combinatorial algorithm based on equilibrium pseudo-flow and penalty functions, extending to minimum cost multi-commodity flow problems.

Findings

Equilibrium pseudo-flow characterizes feasibility of the problem.

Zero-equilibrium pseudo-flow indicates a feasible solution.

The approach can be generalized to minimum cost scenarios.

Abstract

This paper researches combinatorial algorithms for the multi-commodity flow problem. We relax the capacity constraints and introduce a penalty function $h$ for each arc. If the flow exceeds the capacity on arc $a$, arc $a$ would have a penalty cost. Based on the penalty function $h$, a new conception, equilibrium pseudo-flow, is introduced. Then we design a combinatorial algorithm to obtain equilibrium pseudo-flow. If the equilibrium pseudo-flow is a nonzero-equilibrium pseudo-flow, there exists no feasible solution for the multi-commodity flow problem; if the equilibrium pseudo-flow is a zero-equilibrium pseudo-flow, there exists a feasible solution for the multi-commodity flow problem and the zero-equilibrium pseudo-flow is the feasible solution. At last, a non-linear description of the multi-commodity flow problem is given, whose solution is equilibrium pseudo-flow. Besides, the content in this paper can be easily generalized to minimum cost multi-commodity flow problem.