Overcoming Tight Constraints in Soft Happy Colouring

2026-02-16

AI Summary

This work addresses the Soft Happy Coloring (SHC) problem, an NP-hard optimization task that seeks to maximize the number of vertices satisfying a prescribed proportion of same-colored neighbors—a challenge particularly pronounced under tight constraints. The authors propose CE+LS, a novel framework integrating Cross-Entropy (CE) metaheuristics with structure-aware Local Search (LS). By restricting the search space to locally optimal solutions, CE+LS effectively mitigates the probability stagnation commonly observed in CE methods within high-dimensional spaces. This approach achieves, for the first time, provable search space reduction and establishes rigorous convergence guarantees via exponentially decaying KL divergence. Extensive experiments on 28,000 graphs generated from the stochastic block model demonstrate that CE+LS significantly outperforms existing heuristics and membrane algorithms in solution quality, scalability, and performance under stringent constraints.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Search and Optimization: Local Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

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.

Problem

• Soft Happy Colouring
• homophilic networks
• NP-hard optimization
• probabilistic stagnation
• tight constraints

Innovation

• Cross-Entropy method
• Local Search
• Soft Happy Colouring
• Convergence guarantee
• Homophilic networks