Polyhedral structure of the 4-node network design problem

This article studies the polyhedral structure of the 4-node network design problem (NDP). Using a theorem from the previous work of this author, the facets of the 4-node NDP can be translated into facets of larger size problems. The knowledge of complete polyhedral description of the 4-node NDP is important because it implies complete knowledge of 4-partition-based facets of larger NDPs. After reviewing the previously known facets of the 4-node NDP, a new class of facets is derived. An enumerative methodology is presented for determining whether a given set of inequalities provides a complete polyhedral description. By implementing this methodology in a computer code, it is determined that the known facets of the 4-node NDP indeed provide a complete polyhedral description of the problem. Working of the proof methodology is illustrated with examples, and the results of the computer enumeration are reported. © 2009 Wiley Periodicals, Inc. NETWORKS, Vol. 54(3), 139–149 2009