# Exact Algorithms for Arc and Node Routing Problems

Routing problems stand among the hardest combinatorial problems to find high quality bounds or to prove new optimal solutions. In this thesis, we tackle the Capacitated Arc Routing Problem (CARP) and the Generalized Vehicle Routing Problem (GVRP). For both problems, there are a set of customers spread over a given graph, where each customer has a demand which must be serviced by exactly one vehicle from a set of identical vehicles. The traversal costs and a depot vertex are given. The objective is to find routes that collect all the demands, without exceeding the capacity of any vehicle, at minimum cost. For the CARP, the customers are a subset of edges, called the required edges, and for the GVRP, each customer is a subset of vertices, called clusters, where each cluster must be serviced by visiting exactly one vertex of it. Furthermore, it is noteworthy that when every cluster contains just a single vertex, the problem is the Capacitated Vehicle Routing Problem (CVRP). Firstly, we investigate methods to improve lower bounds for large scale instances. We propose to explore the speed of a new dual ascent heuristic to generate capacity cuts. The quality of the cuts found is next improved with a new exact separation which is used in the linear program resolution that follows the dual heuristic. Following, we present a column generation algorithm with an efficient pricing for a special kind of non-elementary routes. The proposed pricing algorithm combines Decremental State-Space Relaxation (DSSR) technique with completion bounds. These techniques allow the strengthening of the domination rule between routes, drastically reducing the total number of labels used during the dynamic programming. Finally, we devise a branch-cut-and-price algorithm which uses the previously presented column generation and cut separation. Moreover, this branch-cut-and-price is implemented using strong branching and reduced cost fixing. At the end of each part, we present computational experiments which evaluate the quality of the proposed algorithms and show new best lower bounds for a large number of CARP and GVRP instances.

## Citation

PhD Thesis, Departamento de Informática, Pontifícia Universidade Católica do Rio de Janeiro, Rio de Janeiro, Brazil, 2012.