A parameterized lower bounding method for the open capacitated arc routing problem

Consider an undirected graph with demands scattered over the edges and a homogeneous fleet of vehicles to service the demands. In the open capacitated arc routing problem (OCARP) the objective is to find a set of routes that collectively service all demands with the minimum cost. Each vehicle has limited capacity and it can start and finish the route at any node. The OCARP is NP-hard, and its applications include meter reading and cutting path determination problems. State-of-the-art solution methods developed for the OCARP are heuristics, which show good tradeoffs between solution quality and processing time, but do not provide optimality certificates of the obtained solutions. This work focuses on a lower bounding method for the OCARP which can be used to better assess the quality of heuristic solutions. We propose the Relaxed Flow method (RF(k)) which involves the resolution of a mixed integer linear formulation where all vehicles' capacities are modeled as flows on an augmented graph. A parameter k controls the model tightness and RF(k) is shown to be at least as tight as the well-known Belenguer and Benavent's formulation for any k > 0. To strengthen the model, capacity cuts were included in RF(k) by means of a branch-and-cut framework. Extensive computational experiments conducted on a set of benchmark instances revealed that our method outperformed previous methods. Computational experiments also demonstrated the importance of the parameterization technique to obtain good results. The previously known lower bounds were improved substantially and optimality certificates were attained in 78.9% of the instances. As far as we know this is the first parameterized lower bounding method proposed for an arc routing problem, and we argue it can be generalized to other variants of arc routing problems and general routing problems.