GRASP Metaheuristics to the Generalized Covering Tour Problem

Let G = (V ∪W,E) be an undirected graph, where V ∪W = {1, ..., n} is the vertex set and E = {(i, j) | i, j ∈ V ∪W, i < j} is the edge set. Vertex s = 1 is the source vertex, V is a set of vertices that might be visited, T ⊆ V is a set of vertices that must be visited (s ∈ T ), and W is a set of vertices that must be covered. A symmetric distance matrix C = (cij), defined on E, uses the Euclidean metric. The Covering Tour Problem, first introduced by Current [1], consists of determining a minimum length tour or a Hamiltonian cycle over a subset of V ∪W in such way that the tour contains all vertices of T , and every vertex of W is covered by the tour, i.e., it lies within a given distance d from a vertex of the tour. The CTP is NP-Hard as it reduces to a TSP when d = 0 and V = W . Such matter has not received much attention in the literature so far. Only two heuristic and two exact methods have been presented for this problem. The first heuristic was proposed to generate a set of solutions to the exact method [2]. The second one [3] combines a heuristic for the TSP to another one for the Set Covering. Maniezzo et al. [4] presented three Scatter Search metaheuristic to the CTP.