An Efficient Local Search for the Minimum Independent Dominating Set Problem

In the present paper, we propose an efficient local search for the minimum independent dominating set problem. We consider a local search that uses $k$-swap as the neighborhood operation. Given a feasible solution $S$, it is the operation of obtaining another feasible solution by dropping exactly $k$ vertices from $S$ and then by adding any number of vertices to it. We show that, when $k=2$, (resp., $k=3$ and a given solution is minimal with respect to 2-swap), we can find an improved solution in the neighborhood or conclude that no such solution exists in $O(nΔ)$ (resp., $O(nΔ^3)$) time, where $n$ denotes the number of vertices and $Δ$ denotes the maximum degree. We develop a metaheuristic algorithm that repeats the proposed local search and the plateau search iteratively, where the plateau search examines solutions of the same size as the current solution that are obtainable by exchanging a solution vertex and a non-solution vertex. The algorithm is so effective that, among 80 DIMACS graphs, it updates the best-known solution size for five graphs and performs as well as existing methods for the remaining graphs.