Constraint Matters: Multi-Modal Representation for Reducing Mixed-Integer Linear programming

Model reduction, which aims to learn a simpler model of the original mixed integer linear programming (MILP), can solve large-scale MILP problems much faster. Most existing model reduction methods are based on variable reduction, which predicts a solution value for a subset of variables. From a dual perspective, constraint reduction that transforms a subset of inequality constraints into equalities can also reduce the complexity of MILP, but has been largely ignored. Therefore, this paper proposes a novel constraint-based model reduction approach for MILPs. Constraint-based MILP reduction has two challenges: 1) which inequality constraints are critical such that reducing them can accelerate MILP solving while preserving feasibility, and 2) how to predict these critical constraints efficiently. To identify critical constraints, we label the tight-constraints at the optimal solution as potential critical constraints and design an information theory-guided heuristic rule to select a subset of critical tight-constraints. Theoretical analyses indicate that our heuristic mechanism effectively identify the constraints most instrumental in reducing the solution space and uncertainty. To learn the critical tight-constraints, we propose a multi-modal representation that integrates information from both instance-level and abstract-level MILP formulations. The experimental results show that, compared to the state-of-the-art MILP solvers, our method improves the quality of the solution by over 50\% and reduces the computation time by 17.47\%.