HyColor: An Efficient Heuristic Algorithm for Graph Coloring
提出一种混合启发式算法HyColor，通过局部决策、图缩减和基于k核与混合度的贪心策略，在大规模稀疏图和小规模稠密图上均优于现有算法，在209个测试实例中194个取得最优解。
The graph coloring problem (GCP) is a classic combinatorial optimization problem that aims to find the minimum number of colors assigned to the vertices of a graph such that no two adjacent vertices receive the same color. GCP has been extensively studied by researchers from various fields, including mathematics, computer science, and biological science. Due to the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {NP}$</tex-math> </inline-formula>-hard nature, many heuristic algorithms have been proposed to solve GCP. However, existing GCP algorithms focus on either small hard graphs or large-scale sparse graphs (with up to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{7}$</tex-math> </inline-formula> vertices). This article presents an efficient hybrid heuristic algorithm for GCP, named <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HyColor</small>, which excels in handling large-scale sparse graphs while achieving impressive results on small dense graphs. The efficiency of <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HyColor</small> comes from the following three aspects: 1) a local decision strategy to improve the lower bound on the chromatic number; 2) a graph-reduction strategy to reduce the working graph; and 3) a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$k$</tex-math> </inline-formula>-core and mixed degree-based greedy heuristic for efficiently coloring graphs. <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HyColor</small> is evaluated against three state-of-the-art GCP algorithms across four benchmarks, comprising three large-scale sparse graph benchmarks and one small dense graph benchmark, totaling 209 instances. The results demonstrate that <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HyColor</small> consistently outperforms existing heuristic algorithms in both solution accuracy and computational efficiency for the majority of instances. Notably, <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HyColor</small> achieved the best solutions in 194 instances (over 93%), with 34 of these solutions significantly surpassing those of other algorithms. Furthermore, <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">HyColor</small> successfully determined the chromatic number and achieved optimal coloring in 128 instances.