Analysis of a Random Local Search Algorithm for Dominating Set

Abstract: Dominating Set is a well-known combinatorial optimization problem which finds application in computational biology or mobile communication. Because of its $\mathrm{NP}$-hardness, one often turns to heuristics for good solutions. Many such heuristics have been empirically tested and perform rather well. However, it is not well understood why their results are so good or even what guarantees they can offer regarding their runtime or the quality of their results. For this, a strong theoretical foundation has to be established. We contribute to this by rigorously analyzing a Random Local Search (RLS) algorithm that aims to find a minimum dominating set on a graph. We consider its performance on cycle graphs with $n$ vertices. We prove an upper bound for the expected runtime until an optimum is found of $\mathcal{O}\left(n^4\log^2(n)\right)$. In doing so, we introduce several models to represent dominating sets on cycles that help us understand how RLS explores the search space to find an optimum. For our proof we use techniques which are already quite popular for the analysis of randomized algorithms. We further apply a special method to analyze a reversible Markov Chain, which arises as a result of our modeling. This method has not yet found wide application in this kind of runtime analysis.