Approximate weighted 3-coloring

The paper considers the NP-hard graph vertex coloring problem, which differs from traditional problems in which it is required to color vertices with a given (or minimal) number of colors so that adjacent vertices have different colors. In the problem under consideration, a simple edge-weighted graph is given. It is required to color its vertices in 3 colors to minimize the total weight of monochromatic (one-color) edges, i.e. edges with the same colors of their end vertices. This problem is poorly investigated. Previously, we developed graph decomposition algorithms that, in particular, allowed us to construct lower bounds for the optimum, as well as several greedy algorithms. In this paper, several new approximation algorithms are proposed. Among them are variable neighborhood search, simulated annealing, genetic algorithm and graph clustering with further finding the optimal coloring in each cluster. A numerical experiment was conducted on random graphs, as well as on real communication graphs. The characteristics of the algorithms are presented both in tables and graphically. The developed algorithms have shown high efficiency.