Visual Evolutionary Optimization on Combinatorial Problems with Multimodal Large Language Models: A Case Study of Influence Maximization

Graph-structured combinatorial problems in complex networks are prevalent in many domains, and are computationally demanding due to their complexity and non-linear nature. Traditional evolutionary algorithms (EAs), while robust, often face obstacles due to content-shallow encoding limitations and lack of structural awareness, necessitating hand-crafted modifications for effective application. In this work, we introduce an original framework, Visual Evolutionary Optimization (VEO), leveraging multimodal large language models (MLLMs) as the backbone evolutionary optimizer in this context. Specifically, we propose a context-aware encoding way, representing the solution of the network as an image. In this manner, we can utilize MLLMs' image processing capabilities to intuitively comprehend network configurations, thus enabling machines to solve these problems in a human-like way. We have developed MLLM-based operators tailored for various evolutionary optimization stages, including initialization, crossover, and mutation. Furthermore, we propose that graph sparsification can effectively enhance the applicability and scalability of VEO on large-scale networks, owing to the scale-free nature of real-world networks. We demonstrate the effectiveness of our method using a well-known task in complex networks, influence maximization, and validate it on eight different real-world networks of various structures. The results have confirmed VEO's reliability and enhanced effectiveness compared to traditional evolutionary optimization.