An intelligent extension of Variable Neighbourhood Search for labelling graph problems

In this paper we describe an extension of the Variable Neighbourhood Search (VNS) which integrates the basic VNS with other complementary approaches from machine learning, statistics and experimental algorithmic, in order to produce high-quality performance and to completely automate the resulting optimization strategy. The resulting intelligent VNS has been successfully applied to a couple of optimization problems where the solution space consists of the subsets of a finite reference set. These problems are the labelled spanning tree and forest problems that are formulated on an undirected labelled graph; a graph where each edge has a label in a finite set of labels L. The problems consist on selecting the subset of labels such that the subgraph generated by these labels has an optimal spanning tree or forest, respectively. These problems have several applications in the real-world, where one aims to ensure connectivity by means of homogeneous connections.

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 and an integer positive value k, and the aim is to find a spanning forest of G having the minimum number of connected components and the upper bound k on the number of labels to use. The problem is related to the minimum labelling spanning tree (MLST) problem, 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 Int-VNS metaheuristic that we propose for the kLSF problem is derived from the promising intelligent VNS strategy recently proposed for the MLST problem, and integrates the basic VNS for the kLSF problem 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.