A Combinatorial Algorithm for the Multi-commodity Flow Problem
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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.
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