Overcoming Tight Constraints in Soft Happy Colouring

TL;DR

This paper introduces CE+LS, a novel algorithm combining probabilistic metaheuristics and local search to effectively solve the NP-hard Soft Happy Colouring problem, especially in challenging tight constraint scenarios.

Contribution

The paper presents CE+LS, a structure-aware local search integrated with the Cross-Entropy method, which overcomes stagnation and guarantees convergence in solving the SHC problem.

Findings

CE+LS outperforms existing heuristics and memetic algorithms.

It demonstrates superior scalability and solution quality on large graphs.

CE+LS remains efficient in tight constraint regimes where others fail.

Abstract

The Soft Happy Colouring (SHC) problem, a mathematical framework for identifying homophilic network structures, seeks to maximise the number of $\rho$-happy vertices, i.e. vertices with at least a proportion $\rho$ of neighbours that share the same colour. Because this NP-hard problem makes exact solutions intractable for large networks, probabilistic metaheuristics such as the Cross-Entropy (CE) method are suitable candidates to be employed. However, pure CE frequently suffers from probabilistic stagnation and non-convergence in high-dimensional spaces. To address this, we introduce {\sf CE+LS}, synergising CE's adaptive learning with a fast, structure-aware local search ({\sf LS}). By restricting the search exclusively to local optima, {\sf CE+LS} learns from high-quality structural characteristics rather than raw random samples. We mathematically prove and empirically demonstrate that this search space reduction resolves CE's stagnation, yielding a strictly convergent algorithm characterised by an exponential decay in Kullback-Leibler divergence. Evaluating {\sf CE+LS} across 28,000 Stochastic Block Model graphs demonstrates that it consistently outperforms existing heuristic and memetic algorithms, exhibiting superior scalability and solution quality. Crucially, {\sf CE+LS} remains highly efficient even in the tight regime, where comparative algorithms fail.