An Efficient Local Search Algorithm for the Minimum $k$ -Dominating Set Problem

The minimum $k$ -dominating set (MKDS) problem, a generalization of the classical minimum dominating set problem, is an important NP-hard combinatorial optimization problem with various applications. First, to alleviate the cycling problem in the local search, a MKDS two-level configuration checking (MKDSCC ² ) strategy is presented. Second, we use the vertex cost scheme to define the scoring mechanism and to improve the solution effectively. Third, by combining MKDSCC ² strategy and the scoring mechanism, we propose a vertex selection strategy to decide which vertex should be added into or removed from the candidate solution. Based on these strategies, an efficient local search algorithm (VSCC ² ), which incorporates a two-level configuration checking strategy, scoring mechanism, and vertex selection strategy, is proposed. We compare the performance of VSCC ² with the classic GRASP algorithm and the famous commercial solver CPLEX on the classical instances. The comprehensive results show that the VSCC ² algorithm is very competitive in terms of solution quality and computing time.