While large language models (LLMs) have emerged as promising tools for CO--either by directly generating solutions or synthesizing solver-specific codes--existing approaches often neglect critical structural priors inherent to CO problems, leading to suboptimality and iterative inefficiency. Inspired by human experts' success in leveraging CO structures for algorithm design, we propose STRCMP, a novel structure-aware LLM-based algorithm discovery framework that systematically integrates structure priors to enhance solution quality and solving efficiency. Our framework combines a graph neural network (GNN) for extracting structural embeddings from CO instances with an LLM conditioned on these embeddings to identify high-performing algorithms in the form of solver-specific codes. This composite architecture ensures syntactic correctness, preserves problem topology, and aligns with natural language objectives, while an evolutionary refinement process iteratively optimizes generated algorithm. Extensive evaluations across Mixed Integer Linear Programming and Boolean Satisfiability problems, using nine benchmark datasets, demonstrate that our proposed STRCMPoutperforms five strong neural and LLM-based methods by a large margin, in terms of both solution optimality and computational efficiency.

Unsupervised learning (UL)-based solvers for combinatorial optimization (CO) train a neural network that generates a soft solution by directly optimizing the CO objective using a continuous relaxation strategy. These solvers offer several advantages over traditional methods and other learning-based methods, particularly for large-scale CO problems. However, UL-based solvers face two practical issues: (I) an optimization issue, where UL-based solvers are easily trapped at local optima, and (II) a rounding issue, where UL-based solvers require artificial post-learning rounding from the continuous space back to the original discrete space, undermining the robustness of the results. This study proposes a Continuous Relaxation Annealing (CRA) strategy, an effective rounding-free learning method for UL-based solvers. CRA introduces a penalty term that dynamically shifts from prioritizing continuous solutions, effectively smoothing the non-convexity of the objective function, to enforcing discreteness, eliminating artificial rounding. Experimental results demonstrate that CRA significantly enhances the performance of UL-based solvers, outperforming existing UL-based solvers and greedy algorithms in complex CO problems. Additionally, CRA effectively eliminates artificial rounding and accelerates the learning process.

We demonstrate that this assumption imposes performance limitations in particular on difficult problem instances.

To this end, we introduce Poppy, a simple training procedure for populations.

Unsupervised learning (UL)-based solvers for combinatorial optimization (CO) train a neural network that generates a soft solution by directly optimizing the CO objective using a continuous relaxation strategy. These solvers offer several advantages over traditional methods and other learning-based methods, particularly for large-scale CO problems.

We introduce Policy Optimization with Multiple Optima (POMO), anend-to-end approach forbuildingsuchaheuristic solver.POMO isapplicable to a wide range of CO problems.