EvoCut: Strengthening Integer Programs via Evolution-Guided Language Models

Integer programming (IP) is central to many combinatorial optimization tasks but remains challenging due to its NP-hard nature. A practical way to improve IP solvers is to manually design acceleration cuts, i.e., inequalities that speed up solving. However, this creative process requires deep expertise and has been difficult to automate. Our proposed framework, EvoCut, automates the generation of acceleration cuts at the symbolic modeling level: it reasons over a symbolic MILP model and a natural language description of the problem to discover a reusable set of acceleration cuts that can be used for each concrete instance of the model. EvoCut (i) initializes a population of candidate cuts via an initializer agent that uses an LLM, (ii) empirically screens candidates on a small verification set by checking that reference solutions remain feasible and that at least one stored LP relaxation solution is cut off, and (iii) iteratively refines the population through evolutionary crossover and mutation agents. Compared to baseline MILP formulations solved with a fixed time budget, EvoCut reduces optimality gaps by up to $76\%$ and reaches target gaps up to $7.2$ times faster (shifted geometric mean speedup). Ablations show its robustness across different LLM backends and across solvers/cut settings. Code: https://github.com/milad1378yz/EvoCut.