Solving the Minimum Dominating Set Problem of Partitioned Graphs Using a Hybrid Bat Algorithm

Abstract

The minimum dominating set (MDS) is the problem of finding the minimum number of nodes that have connections to all other nodes in a given graph. This problem belongs to the NP-complete complexity class that cannot be solved exactly in a polynomial time. Hence, we deployed a stochastic method to estimate good solutions for the MDS in a reasonable time. However, the problem space of the MDS problem grows exponentially with respect to the graph size. Therefore, our proposed method partitions the given graph to sub-graphs that can be tackled independently, which reduces the computational time of finding the MDS solution. This paper investigates the swarm intelligence behaviour represented by a population-based approach called the bat algorithm (BA) to find the smallest set of nodes that dominate the graph. The BA explores a wide area of the search space; thus, it is capable of the diversification procedure. However, population-based algorithms are not good in exploiting the search space in comparison to single-solution based methods, therefore, we included the Simulated annealing (SA) algorithm to balance between exploitation and exploration in order to reach a best possible solution. To analyse the performance of the proposed partitioning scheme, we experimented with the hybrid algorithm on the graph, with and without partitioning. The gained results showed significant speed up when the partitioning scheme was applied.