Approximate weighted 3-coloring

TL;DR

This paper introduces new approximation algorithms for the NP-hard weighted 3-coloring problem, demonstrating their high efficiency through experiments on random and real graphs.

Contribution

It proposes several novel approximation algorithms, including variable neighborhood search, simulated annealing, genetic algorithm, and graph clustering methods.

Findings

Algorithms show high efficiency in experiments

Effective on both random and real communication graphs

Provides detailed performance analysis in tables and graphs

Abstract

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.