On the Effectiveness of the Linear Programming Relaxation of the 0-1 Multi-commodity Minimum Cost Network Flow Problem
Several studies have reported that the linear program relaxation of integer multi-commodity network flow problems often provides integer optimal solutions. We explore this phenomenon with a 0-1 multi-commodity network with mutual arc capacity constraints. Characteristics of basic solutions in the linear programming relaxation problem of the 0-1 multi-commodity problem are identified. Specifically, necessary conditions for a linear programming relaxation to have a non-integer solution are presented. Based on the observed characteristics, a simple illustrative example problem is constructed to show that its LP relaxation problem has integer optimal solutions with a relatively high probability. Furthermore, to investigate whether or not and under what conditions this tendency applies to large-sized problems, we have carried out computational experiments by using randomly generated problem instances. The results of our computational experiment indicate that there exists a narrow band of arc density in which the 0-1 multi-commodity problems possess no integer optimal solutions.
KeywordsLinear Program Relaxation Basic Feasible Solution Vehicle Schedule Integer Optimal Solution Multicommodity Network
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- 12.Ozdaglar, A.E., Bertsekas, D.P.: Optimal Solution of Integer Multicommodity Flow Problems with Application in Optical Networks. In: Proc. of Symposium on Global Optimization, Santorini, Greece (2003)Google Scholar