On the Generalized Arc Routing Problem

In this work we study the Generalized Arc Routing Problem (GARP) on an undirected graph. In the GARP a given graph is partitioned into clusters and a route is sought that traverses at least one edge of each cluster. Broadly speaking, the GARP is the arc routing counterpart of the Generalized Traveling Salesman Problem (GTSP). In the GTSP the set of vertices of a given graph is partitioned into clusters and a route is sought that visits at least one vertex of each cluster. Different versions of the GTSP have been studied by various authors (see, for instance, Noon and Bean, 1991; Laporte et al., 1987; Fischetti et al., 1995; Baldacci et al., 2010; Cacchiani et al., 2011). Potential applications of the GARP arise in different contexts. In meter reading, one of the classical applications of arc routing, current technology allows to traverse just a few of the edges where meters are placed. Also in quality control for networks maintenance in which only a small subset of the edges of a network has to be traversed. Furthermore, the GARP is most appropriate for modeling location/arc-routing problems in which, due to their characteristics, facilities cannot be located at the nodes of a network, so they have to be located at the edges of some given areas (clusters) and a route connecting the facilities is sought. We introduce the GARP as a single vehicle arc routing problem on an undirected graph, and give a first linear integer formulation using two sets of binary variables, in the spirit of current formulations for this type of problems. After studying some optimality conditions, a tighter formulation with only one set of binary variables is proposed. The polyhedron associated with the latter formulation is studied and some facets and families of valid inequalities are given. In particular, we present two new families of inequalities which are valid for the GARP and extend the well-known co-circuit and matching inequalities, and we establish some relationship between them. We also study the separation problem for the different families of valid inequalities, and we propose a solution algorithm which iteratively reinforces the current LP relaxation by incorporating separated inequalities. Finally, we report on the numerical results of a series of computational experiments.