The Capacitated Arc Routing Problem: Valid Inequalities and Facets

In this paper we study the polyhedron associated with the Capacitated Arc Routing Problem (CARP) where a maximum number K of vehicles is available. We show that a subset of the facets of the CARP polyhedron depends only on the demands of the required edges and they can be derived from the study of the Generalized Assignment Problem (GAP). The conditions for a larger class of valid inequalities to define facets of the CARP polyhedron still depend on the properties of the GAP polyhedron. We introduce the special case of the CARP where all the required edges have unit demand (CARPUD) to avoid the number problem represented by the GAP. This allows us to make a polyhedral study in which the conditions for the inequalities to be facet inducing are easily verifiable. We give necessary and sufficient conditions for a variety of inequalities, which are valid for CARP, to be facet inducing for CARPUD.The resulting partial description of the polyhedron has been used to develop a cutting plane algorithm for the Capacitated Arc Routing Problem. The lower bound provided by this algorithm outperformed all the existing lower bounds for the CARP on a set of 34 instances taken from the literature.