Behavior and Representation in Large Language Models for Combinatorial Optimization: From Feature Extraction to Algorithm Selection

Recent advances in Large Language Models (LLMs) have opened new perspectives for automation in optimization. While several studies have explored how LLMs can generate or solve optimization models, far less is understood about what these models actually learn regarding problem structure or algorithmic behavior. This study investigates how LLMs internally represent combinatorial optimization problems and whether such representations can support downstream decision tasks. We adopt a twofold methodology combining direct querying, which assesses LLM capacity to explicitly extract instance features, with probing analyses that examine whether such information is implicitly encoded within their hidden layers. The probing framework is further extended to a per-instance algorithm selection task, evaluating whether LLM-derived representations can predict the best-performing solver. Experiments span four benchmark problems and three instance representations. Results show that LLMs exhibit moderate ability to recover feature information from problem instances, either through direct querying or probing. Notably, the predictive power of LLM hidden-layer representations proves comparable to that achieved through traditional feature extraction, suggesting that LLMs capture meaningful structural information relevant to optimization performance.