Overcoming tight constraints in soft happy colouring

The Soft Happy Colouring (SHC) problem, a mathematical framework for identifying homophilic network structures, seeks to maximise the number of -happy vertices, i.e., vertices with at least a proportion of neighbours that share the same colour. Because this NP-hard problem makes finding exact solutions intractable for large networks, probabilistic metaheuristics such as the Cross-Entropy (CE) method are suitable candidates. However, pure CE frequently suffers from stagnation of the probability distributions and non-convergence in high-dimensional spaces. To address this, we introduce CE+LS , synergising CE’s adaptive learning with a fast, structure-aware local search ( LS ). By restricting the search exclusively to local optima, CE+LS learns from high-quality structural characteristics rather than raw random samples. We mathematically and empirically demonstrate that this search space reduction resolves CE’s stagnation, yielding a convergent algorithm. Evaluating CE+LS across 28,000 Stochastic Block Model graphs, validated by non-parametric statistical testing, demonstrates that it consistently outperforms existing heuristic and memetic algorithms. Furthermore, benchmarking against the commercial exact solver, CPLEX, on real-world networks confirms that CE+LS identifies near-optimal configurations in a fraction of the required computational time for CPLEX. Crucially, CE+LS remains highly efficient even in the tight constraint regime, where comparative algorithms usually fail.