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.

翻译：支配集是一个著名的组合优化问题，在计算生物学和移动通信等领域具有重要应用。由于其$\mathrm{NP}$难特性，研究者通常采用启发式算法来获得近似解。许多此类启发式算法经过实证检验表现出良好性能，但其结果优异的原因、运行时间保证以及解的质量保证尚未得到充分理解。为此，需要建立坚实的理论基础。本文通过严格分析一种旨在寻找图的最小支配集的随机局部搜索（RLS）算法，为此领域作出贡献。我们研究了该算法在具有$n$个顶点的环图上的性能，证明了其找到最优解的期望运行时间上界为$\mathcal{O}\left(n^4\log^2(n)\right)$。在分析过程中，我们引入了多种表示环上支配集的模型，这些模型有助于理解RLS如何探索搜索空间以找到最优解。证明过程中采用了随机算法分析中已广泛使用的技术，并特别应用了一种分析可逆马尔可夫链的方法——该马尔可夫链由我们的建模过程产生，此类方法在当前运行时间分析领域中尚未得到广泛应用。