Approximate weighted 3-coloring
Adil Erzin, Roman Plotnikov, Georgii Zhukov
TL;DR
The paper tackles the NP-hard problem of 3-coloring a weighted graph to minimize the total weight of monochromatic edges, formalized with binary variables $x_i^k$ and $y_{ij}^k$ and linear constraints. It introduces spectral clustering based graph decomposition using the normalized Laplacian $L_{sym}$ to partition the graph into subgraphs and enable exact coloring within clusters, together with five meta-heuristics: Iterative Partial Improvements, Hybrid Simulated Annealing, Variable Neighborhood Search, Genetic Local Search, and AllMetaheuristics. Two lower bounds, LB from $\text{Erzin:24}$ and LB2 from spectral clustering, are used to gauge solution quality; experiments on real wireless networks, random graphs, and unit-disk graphs show that GLS often yields the best solutions and that AllMH is robust across problems. The results highlight the practical effectiveness of combining spectral decomposition with diverse meta-heuristics for scalable, high-quality approximations in weighted graph 3-coloring.
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.