Mixed-Integer Linear Programming (MILP) is a cornerstone of combinatorial optimization, yet solving large-scale instances remains a significant computational challenge. Recently, Graph Neural Networks (GNNs) have shown promise in accelerating MILP solvers by predicting high-quality solutions. However, we identify that existing methods misalign with the intrinsic structure of MILP problems at two levels. At the leaning objective level, the Binary Cross-Entropy (BCE) loss treats variables independently, neglecting their relative priority and yielding plausible logits. At the model architecture level, standard GNN message passing inherently smooths the representations across variables, msking the natural competitive relationships within constraints. To address these challenges, we propose CoCo-MILP, which explicitly models inter-variable Contrast and intra-constraint Competition for advanced MILP solution prediction. At the objective level, CoCo-MILP introduces the Inter-Variable Contrastive Loss (VCL), which explicitly maximizes the embedding margin between variables assigned one versus zero. At the architectural level, we design an Intra-Constraint Competitive GNN layer that, instead of homogenizing features, learns to differentiate representations of competing variables within a constraint, capturing their exclusionary nature. Experimental results on standard benchmarks demonstrate that CoCo-MILP significantly outperforms existing learning-based approaches, reducing the solution gap by up to 68.12% compared to traditional solvers.

Large Language Model (LLM)-based optimization has recently shown promise for autonomous problem solving, yet most approaches still cast LLMs as passive constraint checkers rather than proactive strategy designers, limiting their effectiveness on complex Constraint Optimization Problems (COPs).To address this, we present AutoCO, an end-to-end Automated Constraint Optimization method that tightly couples operations-research principles of constraint relaxation with LLM reasoning.A core innovation is a unified triplerepresentation that binds relaxation strategies, algorithmic principles, and executable codes.This design enables the LLM to synthesize, justify, and instantiate relaxation strategies that are both principled and executable.To navigate fragmented solution spaces, AutoCO employs a bidirectional global-local coevolution mechanism, synergistically coupling Monte Carlo Tree Search (MCTS) for global relaxationtrajectory exploration with Evolutionary Algorithms (EAs) for local solution intensification.This continuous exchange of priors and feedback explicitly balances diversification and intensification, thus preventing premature convergence.Extensive experiments on three challenging COP benchmarks validate AutoCO's consistent effectiveness and superior performance, especially in hard regimes where current methods degrade.Results highlight Au-toCO as a principled and effective path toward proactive, verifiable LLM-driven optimization.

Mixed-Integer Linear Programming (MILP) is a fundamental and powerful framework for modeling complex optimization problems across diverse domains. Recently, learning-based methods have shown great promise in accelerating MILP solvers by predicting high-quality solutions. However, most existing approaches are developed and evaluated in single-domain settings, limiting their ability to generalize to unseen problem distributions. This limitation poses a major obstacle to building scalable and general-purpose learning-augmented solvers. To address this challenge, we introduce RoME, a domain-Robust Mixture-of-Experts (MoE) framework for predicting MILP solutions across domains. RoME dynamically routes problem instances to specialized experts based on learned task embeddings. The model is trained using a two-level distributionally robust optimization strategy: inter-domain to mitigate global shifts across domains, and intra-domain to enhance local robustness by introducing perturbations on task embeddings. We reveal that cross-domain training not only enhances the model's generalization capability to unseen domains but also improves performancewithin each individual domain by encouraging the model to capture more general intrinsic combinatorial patterns. Specifically, a single RoME model trained on three domains achieves an average improvement of $67.7\%$ then evaluated on five diverse domains. We further test the pretrained model on MIPLIB in a zero-shot setting, demonstrating its ability to deliver measurable performance gains on challenging real-world instances where existing learning-based approaches often struggle to generalize.

Many complex problems encountered in both production and daily life can be conceptualized as combinatorial optimization problems (COPs). Many ad-hoc deep learning methods have been proposed to solve these problems, but there still lacks an effective unified framework. In this work, we take a step towards this goal by designing an unified end-to-end deep reinforcement learning framework named CONQUER. CONQUER first translates various COPs into the unified QUBO (Quadratic Unconstrained Binary Optimization) formalization, and afterwards trains a neural QUBO solver to search high-quality strategies, which are then checked and repaired to obtain the solutions for original COPs. The QUBO solver is a purely data-driven neural model, which does not rely on any expert inputs. We adopt the graph transformer network to represent the QUBO graph, so as to effectively capture the features of problem objective and constraints. The Deep Q-network is utilized to train the QUBO solver to find long-sighted solution strategies. Despite being trained solely on small synthetic graphs, CONQUER exhibits promising generality on much larger instances, and can be easily adapted to real-world datasets. Experimentally, we show that CONQUER achieves the state-of-the-art performances on four COPs over graphs, including minimum vertex cover, maximum independent set, maximum clique and maximum cut problem.

The combinatorial optimization problems (COPs) over graph are of great significance both in theory and practice, covering a wide range of scenarios in daily life and industrial production. Recent years, reinforcement learning (RL) based models have emerged as a promising direction, which treat solving the COPs as a heuristic learning problem. However, current finite-horizon Markov Decision Process (MDP) based RL models are not allowed to explore adquately for improving solutions at test time, which may be necessary given the complexity of NP-hard optimization tasks. Some recent attempts solve this issue by focusing on reward design and state feature engineering, which are tedious and ad-hoc. To address this challenge, we introduce a physics-inspired technique called gauge transformation (GT), which is highly effective in enabling RL agents to explore and continuously enhance solution quality during testing. GT seamlessly transforms any state within the MDP back to its initial state, allowing the RL agent to continue exploration within the transformed space. Empirically, we demonstrate that traditional RL models equipped with the GT technique achieve the SOTA performance on the MaxCut problem. Moreover, GT is exclusively applied during testing and does not alter the training phase of the model. It can be readily integrated into existing RL models, providing a pathway for more effective exploration in solving the COPs.