Structure-Aware Cooperative Ensemble Evolutionary Optimization on Combinatorial Problems with Multimodal Large Language Models

Evolutionary algorithms (EAs) have proven effective in exploring the vast solution spaces typical of graph-structured combinatorial problems. However, traditional encoding schemes, such as binary or numerical representations, often fail to straightforwardly capture the intricate structural properties of networks. Through employing the image-based encoding to preserve topological context, this study utilizes multimodal large language models (MLLMs) as evolutionary operators to facilitate structure-aware optimization over graph data. To address the visual clutter inherent in large-scale network visualizations, we leverage graph sparsification techniques to simplify structures while maintaining essential structural features. To further improve robustness and mitigate bias from different sparsification views, we propose a cooperative evolutionary optimization framework that facilitates cross-domain knowledge transfer and unifies multiple sparsified variants of diverse structures. Additionally, recognizing the sensitivity of MLLMs to network layout, we introduce an ensemble strategy that aggregates outputs from various layout configurations through consensus voting. Finally, experiments on real-world networks through various tasks demonstrate that our approach improves both the quality and reliability of solutions in MLLM-driven evolutionary optimization.