Hypergraph Neural Networks Accelerate MUS Enumeration

Graph neural networks (GNNs) are increasingly applied to hard optimization problems, often claiming superiority over classical heuristics. However, such claims risk being unsolid due to a lack of standard benchmarks on truly hard instances. From a statistical physics perspective, we propose new hard benchmarks based on random problems. We provide these benchmarks, along with performance results from both classical heuristics and GNNs. Our fair comparison shows that classical algorithms still outperform GNNs. We discuss the challenges for neural networks in this domain. Future claims of superiority can be made more robust using our benchmarks, available at https://github.com/ArtLabBocconi/RandCSPBench.

We propose Tiny Recursive Models for Combinatorial Optimization (\ours{}), a general neural method for combinatorial optimization that scales both depth (how often we recursively invoke our network) and width (how much we sample in parallel). Both are fundamental for combinatorial optimization: hard instances demand a large amount of compute, while a small network is essential to avoid overfitting and capture the algorithmic essence of optimization. In particular, our method consists of a graph-aware tiny recursive model that iterates on a latent state with adaptive halting and needs only a lightweight problem-specific decoder. Compared with previous heatmap-based general neural solvers, it achieves a better balance between solution quality and inference speed on both the Traveling Salesman Problem~(TSP) and the Maximum Independent Set~(MIS) problem, and remains competitive with hybrid methods that combine neural components with heuristics specific to each problem. With the same backbone architecture for both tasks, \ours{} outperforms every diffusion-based solver on TSP from 500 to 10,000 cities at a lower inference cost, and on the standard Erdős--Rényi-[700-800] MIS benchmark it surpasses all neural solvers except those that only work well on MIS. We then explore self-relabeling for self-supervised training. We periodically replace the current set of training labels with the model's own better solutions, as an alternative training signal. Self-relabeling can, while forgoing supervision from near-optimal solutions, still result in on-par quality.

Foundational optimization embeddings have recently emerged as powerful pre-trained representations for mixed-integer programming (MIP) problems. These embeddings were shown to enable cross-domain transfer and reduce reliance on solver-generated labels. In this work, we investigate whether such representations generalize beyond optimization to decision problems, focusing on Boolean satisfiability (SAT). We adapt the foundational optimization architecture to SAT by mapping CNF formulas into the same bipartite constraint-variable graph representation used for MIPs. This allows direct reuse of the pre-trained embedding model without architectural changes or supervised fine-tuning. Our results show that these embeddings capture structural regularities in SAT instances and support unsupervised tasks such as instance clustering and distribution identification. We demonstrate, for the first time, that foundational optimization embeddings can transfer to constraint satisfaction domains. Our findings is a step toward a unified representational framework for both optimization and decision problems.