A new local search algorithm with greedy crossover restart for the dominating tree problem

The dominating tree problem (DTP), a variant of the classical minimum dominating set problem, aims to find a dominating tree of minimum costs on a given connected, undirected and edge-weighted graph. In this paper, we design an efficient local search algorithm named LSGCR_DTP to solve DTP. To this end, we propose four new strategies to make it efficient. Firstly, we design a new connecting method for non-connected dominating sets, which no longer depends on the shortest paths between each pair of vertices used in the existing DTP solvers. Secondly, we propose a new method for improving the current feasible connected dominating sets by adding some special vertices to them. Thirdly, a vertex selection strategy for balancing the connecting property and dominating property of solution is introduced. Fourthly, a new restart strategy based on greedy mechanism and crossover operation is integrated to our local search algorithm. Experimental results show that the LSGCR_DTP algorithm can find the best solutions on about 81.5% conventional benchmarks, which outperforms state-of-the-art DTP solvers, including O_ABCDT, EA/G, ABC_DTP and GAITLS. Specially, the solution records of 7 instances are broken by LSGCR_DTP, and LSGCR_DTP can be used to solve many massive graphs with tens or even hundreds of thousand vertices which cannot be solved by the contrastive algorithm.