A Combinatorial Algorithm for the Multi-commodity Flow Problem

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.