HyColor: An Efficient Heuristic Algorithm for Graph Coloring

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 theNP-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 to10⁷vertices). This article presents an efficient hybrid heuristic algorithm for GCP, namedHyColor, which excels in handling large-scale sparse graphs while achieving impressive results on small dense graphs. The efficiency ofHyColorcomes 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) ak-core and mixed degree-based greedy heuristic for efficiently coloring graphs.HyColoris 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 thatHyColorconsistently outperforms existing heuristic algorithms in both solution accuracy and computational efficiency for the majority of instances. Notably,HyColorachieved the best solutions in 194 instances (over 93%), with 34 of these solutions significantly surpassing those of other algorithms. Furthermore,HyColorsuccessfully determined the chromatic number and achieved optimal coloring in 128 instances.