Variable neighborhood search for solving the k-domination problem

In this paper we are concerned with solving a generalized version of the well-known minimum dominating set problem, the so-called k-domination problem, k ∈ ℕ. This problem is about finding a minimal cardinality subset D of vertices of a graph G = (V, E) such that every υ ∈ V belongs to D or has at least k neighbors from D. The k-domination problem has applications in distributed systems, biological networks etc. We propose a variable neighborhood search (VNS) metaheuristic for solving the k-domination problem. The Vns is equipped with an efficient fitness function that allows it to consider both feasible and infeasible solutions, while appropriately penalizing infeasible solutions. The control parameters of the Vns are tuned using a grid search approach. The method is compared to the best known heuristic approaches from the literature: the beam search and several greedy approaches. Experimental evaluations are performed on a real-world benchmark set whose instances represent the road networks of different cities. The Vns provided new state-of-the-art results for all considered problem instances with k ∈ {1, 2, 4}.