The open Capacitated Arc Routing Problem: Complexity and algorithms

The Capacitated Arc Routing Problem (CARP) [1] is a well-known combinatorial optimization problem in which, given an undirected graph G(V ;E) with non-negative costs and demands associated to the edges, we have M identical vehicles with capacity D that must traverse all edges with positive demand. The vehicles must start and finish their tours at a depot node, without transgressing their capacity. The objective is to search for a solution of minimum cost. The CARP was shown to be NP-hard [1], which means that an exact polynomial algorithm for this problem is most unlikely. Nevertheless, there are several heuristics that tackle this problem and which perform very well in most cases. Some of these are path-scanning [2], [3], augment-merge [1], augment-insert [4], among other heuristics [5], [6]. Even better solutions were obtained through meta-heuristics such as the tabu search [7], as well as a genetic algorithm [8], a hybrid tabu-scatter search algorithm [9], and a guided local search [10]. There is also an exact algorithm for the CARP based upon a branch-and-bound strategy [11] which, however, can solve only small size instances (up to 20 required edges). Furthermore, there are algorithms that can determine upper and lower bounds for the CARP [12].