Towards an intelligent VNS heuristic for the k-labelled spanning forest problem

In a currently ongoing project, we investigate a new possibility for solving the k-labelled spanning forest (kLSF) problem by an intelligent Variable Neighbourhood Search (Int-VNS) metaheuristic. In the kLSF problem we are given an undirected input graph G = (V, E, L) where V is the set of nodes, E the set of edges, that are labelled on the set L of labels, and an integer positive value ¯ k, and the aim is to find a spanning forest G * = (V, E * , L *) of the input graph having the minimum number of connected components, i.e. min|Comp(G *)|, and the upper bound ¯ k on the number of labels to use, |L * | ≤ ¯ k. The problem is related to the minimum labelling spanning tree (MLST) problem [1], whose goal is to get the spanning tree of the input graph with the minimum number of labels, and has several applications in the real-world, where one aims to ensure connectivity by means of homogeneous connections. The kLSF problem was recently introduced in [2] along with the proof of its NP-hardness; therefore any practical solution approach requires heuristics [2, 3]. In particular our aim is to present an intelligent VNS which is aimed to achieve further improvements for the kLSF problem. This approach is derived from the promising strategy recently proposed in [4] for the MLST problem, and integrates the basic VNS for the kLSF problem in [3] with other complementary approaches from machine learning, statistics and experimental algorithmics, in order to produce high-quality performance and to completely automate the resulting strategy. The first extension that we introduce is a local search mechanism that is inserted at top of the basic VNS. The resulting local search method is referred to as Complementary Variable Neighbourhood Search (Co-VNS) [4]. selected from the complementary space of L * , referred to as Co space , defined as the set of all the labels that are not contained in L * , that is L∆L *. The iterative process of extraction of a new solution from the complementary space of the current solution helps to escape the algorithm from possible traps in local minima, since the complementary solution lies in a very different zone of the search space with respect to the incumbent solution. Successively, the basic VNS [3] is applied in order to improve the resulting solution. At the …