CoCo-MILP: Inter-Variable Contrastive and Intra-Constraint Competitive MILP Solution Prediction

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, missing 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. Our code is available at https://github.com/happypu326/CoCo-MILP.

Transform then Explore: a Simple and Effective Technique for Exploratory Combinatorial Optimization with Reinforcement Learning

Many complex problems encountered in both production and daily life can be conceptualized as combinatorial optimization problems (COPs) over graphs. Recent years, reinforcement learning (RL) based models have emerged as a promising direction, which treat the COPs solving as a heuristic learning problem. However, current finite-horizon-MDP based RL models have inherent limitations. They 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. In this work, we instead propose a much simpler but more effective technique, named gauge transformation (GT). The technique is originated from physics, but is very effective in enabling RL agents to explore to continuously improve the solutions during test. Morever, GT is very simple, which can be implemented with less than 10 lines of Python codes, and can be applied to a vast majority of RL models. Experimentally, we show that traditional RL models with GT technique produce the state-of-the-art performances on the MaxCut problem. Furthermore, since GT is independent of any RL models, it can be seamlessly integrated into various RL frameworks, paving the way of these models for more effective explorations in the solving of general COPs.