Local search is a well-known heuristic method used in optimization. In this thesis, we explore its capabilities on the vertex coloring problem, an $NP$-hard problem with relevance in both theoretical analysis and practical application. To recognize limitations in the applicability of local search of the vertex coloring problem, we analyze local search landscapes on differently-structured bipartite graphs. We identify structures that ensure only global optima can exist as well as ones that enable the existence of non-global local optima, showing that on general bipartite graphs, it is possible for local search to return arbitrarily bad results. Further, we analyze the capabilities of local search on graphs where a local optimum can be found. To do so, we introduce a gray-box local search mutation operator that removes less frequent colors with higher probability and prove that it finds an optimal coloring on complete bipartite graphs in an expected run time of $Θ(n \log n)$. This is a drastic improvement to the exponential tun time of the black-box Random Local Search, showing that gray-box mutation operators can improve the run time of local search.

翻译：局部搜索是一种著名的优化启发式方法。本文探讨了其在顶点着色问题上的能力，该问题是一个具有理论分析与实际应用双重意义的NP难问题。为认识局部搜索在顶点着色问题上适用性的局限性，我们分析了不同结构二分图上的局部搜索景观。我们识别出既能确保仅存在全局最优解、又能允许非全局局部最优解存在的结构，表明在一般二分图上，局部搜索可能返回任意差的结果。进一步，我们分析了局部搜索在可找到局部最优解的图上的能力。为此，我们引入了一种灰盒局部搜索变异算子，该算子以更高概率移除出现频率较低的色彩，并证明其在完全二分图上能在期望运行时间Θ(n log n)内找到最优着色。这较之黑盒随机局部搜索的指数级运行时间有显著改进，表明灰盒变异算子能够提升局部搜索的运行时间。