Given a graph $G=(V,E)$, the $b$-coloring problem consists in attributing a color to every vertex in $V$ such that adjacent vertices receive different colors, every color has a $b$-vertex, and the number of colors is maximized. A $b$-vertex is a vertex adjacent to vertices colored with all used colors but its own. The $b$-coloring problem is known to be NP-Hard and its optimal solution determines the $b$-chromatic number of $G$, denoted $\chi_b(G)$. This paper presents an integer programming formulation and a very effective multi-greedy randomized heuristic which can be used in a multi-start metaheuristic. In addition, a matheuristic approach is proposed combining the multi-start multi-greedy randomized metaheuristic with a MIP (mixed integer programming) based local search procedure using the integer programming formulation. Computational experiments establish the proposed multi-start metaheuristic as very effective in generating high quality solutions, along with the matheuristic approach successfully improving several of those results. Moreover, the computational results show that the multi-start metaheuristic outperforms a state-of-the-art hybrid evolutionary metaheuristic for a subset of the large instances which were previously considered in the literature. An additional contribution of this work is the proposal of a benchmark instance set, which consists of newly generated instances as well as others available in the literature for classical graph problems, with the aim of standardizing computational comparisons of approaches for the $b$-coloring problem in future works.

翻译：鉴于一个GG=(V,E),美元彩色问题在于将一个彩色问题归结到美元每面顶点上,使相邻的顶点得到不同的颜色,每个颜色都有美元顶点,颜色的数量最大化。一个美元顶点是一个与所有用过的颜色而用自己的颜色来颜色的顶点相邻的顶点。已知美元彩色问题是NP-Hard,其最佳解决方案决定了美元每色的色度比较数,即$G$,意指$ch_b(G)美元。本文展示了一个整形的编程配方,每个颜色都有一个非常有效的多色点随机化的色点数。此外,还提出了一种数学方法,将多色点的多色点随机化美度与以 MIP (混合整数编程编程)为基础的本地搜索程序结合起来,用整数位版编程格式来决定了美元、记号$$@chi_b(G)的色比较。本文展示了一个整数级编程中的拟议多色进点计算公式,在开始的基点计算结果中成功地展示了数学结果。