Enhancing Soft Happiness via Evolutionary Algorithms

2025-08-28

AI Summary

This paper addresses the Soft ρ-Happy Coloring problem: given a partially colored graph, find a complete proper coloring that maximizes the number of ρ-happy vertices—those for which at least a ρ-fraction of their neighbors share the same color. As an NP-hard optimization problem, it bridges combinatorial optimization and network analysis. We propose a novel evolutionary framework integrating genetic algorithms with gene-like operators, augmented by a Local Maximal Colouring initialization strategy and a multi-stage local search mechanism to enhance both initial population quality and local exploitation capability. Experiments on random graph benchmarks demonstrate that our approach achieves a higher count of ρ-happy vertices, attains superior community structure identification accuracy compared to state-of-the-art methods, and successfully computes more feasible complete colorings. These results validate the effectiveness and advancement of our method for joint combinatorial optimization and network analysis tasks.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

For $0leq ρleq 1$, a $ρ$-happy vertex $v$ in a coloured graph shares colour with at least $ρmathrm{deg}(v)$ of its neighbours. Soft happy colouring of a graph $G$ with $k$ colours extends a partial $k$-colouring to a complete vertex $k$-colouring such that the number of $ρ$-happy vertices is maximum among all such colouring extensions. The problem is known to be NP-hard, and an optimal solution has a direct relation with the community structure of the graph. In addition, some heuristics and local search algorithms, such as {sf Local Maximal Colouring} ({sf LMC}) and {sf Local Search} ({sf LS}), have already been introduced in the literature. In this paper, we design Genetic and Memetic Algorithms for soft happy colouring and test them for a large set of randomly generated partially coloured graphs. Memetic Algorithms yield a higher number of $ρ$-happy vertices, but Genetic Algorithms can perform well only when their initial populations are locally improved by {sf LMC} or {sf LS}. Statistically significant results indicate that both Genetic and Memetic Algorithms achieve high average accuracy in community detection when their initial populations are enhanced using {sf LMC}. Moreover, among the competing methods, the evolutionary algorithms identified the greatest number of complete solutions.

Problem

• Maximizing ρ-happy vertices in graph coloring extensions
• Solving NP-hard soft happy coloring via evolutionary algorithms
• Enhancing community detection accuracy through optimized coloring

Innovation

• Genetic Algorithms for soft happy coloring optimization
• Memetic Algorithms combining evolution with local search
• Initial population enhancement using Local Maximal Coloring