A variable neighborhood search and simulated annealing hybrid for the profile minimization problem

Given an undirected simple graph G, the profile minimization problem (PMP) is to find an ordering of the vertices of the graph G such that the sum of the profiles of all its vertices is minimized. The profile of the vertex v in position i is defined as max{0,i−hv}, where hv is the position of the leftmost vertex among all vertices adjacent to v in G. We propose an approach for the PMP, which combines a variable neighborhood search (VNS) scheme with the multi-start simulated annealing (MSA) technique. The solution delivered by MSA is submitted as input to the VNS component of the method. The VNS algorithm heavily relies on a fast insertion neighborhood exploration procedure. We show that the time complexity of this procedure is O(n2), where n is the order of G. We have found empirically that it is advantageous to give between 50 and 75% of the computation time to MSA and the rest to VNS. The results of the computational experiments demonstrate the superiority of our MSA-VNS algorithm over the current state-of-the-art metaheuristic approaches for the PMP. Using MSA-VNS, we improved the best known solutions for 50 well-recognized benchmark PMP instances in the literature. The source code implementing MSA-VNS is made publicly available as a benchmark for future comparisons.