Application of CMSA to the Minimum Positive Influence Dominating Set Problem

Construct, Merge, Solve & Adapt (CMSA) is a recently developed algorithm for solving combinatorial optimization problems. It combines heuristic elements, such as the probabilistic generation of solutions, with an exact solver that is iteratively applied to sub-instances of the tackled problem instance. In this paper, we present the application of CMSA to an NP-hard problem from the family of dominating set problems in undirected graphs. More specifically, the application in this paper concerns the minimum positive influence dominating set problem, which has applications in social networks. The obtained results show that CMSA outperforms the current state-of-the-art metaheuristics from the literature. Moreover, when instances of small and medium size are concerned CMSA finds many of the optimal solutions provided by CPLEX, while it clearly outperforms CPLEX in the context of the four largest, respectively more complicated, problem instances.

On Eternal Domination of Generalized J s , m

An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only. These guards can defend any infinite series of attacks, an attack is defended by moving one guard along an edge from its position to the attacked vertex. We consider the “all guards move” of the eternal dominating set problem, in which one guard has to move to the attacked vertex, and all the remaining guards are allowed to move to an adjacent vertex or stay in their current positions after each attack in order to form a dominating set on the graph and at each step can be moved after each attack. The “all guards move model” is called the m -eternal domination model. The size of the smallest m -eternal dominating set is called the m -eternal domination number and is denoted by γ m ∞ G . In this paper, we find the domination number of Jahangir graph J s , m for s ≡ 1 , 2 mod 3 , and the m -eternal domination numbers of J s , m for s , m are arbitraries.