{"ID":3050057,"CreatedAt":"2026-06-04T02:13:16.786527022Z","UpdatedAt":"2026-06-06T11:59:53.540122282Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2606.04816","arxiv_id":"2606.04816","title":"Beyond Objective Equivalence: Constraint Injection for LLM-Based Optimization Modeling on Vehicle Routing Problems","abstract":"Large language models (LLMs) increasingly translate natural-language optimization problems into executable solver code. Yet for constraint-dense operations research (OR) problems, existing data-filtering and training pipelines largely rely on objective-equivalence signals such as differential testing and answer agreement, which a program can pass while adding spurious constraints or silently omitting required ones, whenever those constraints are non-binding on the tested instance. We propose constraint injection, which uses feasible probes to expose spurious over-constraint and one-constraint-violating probes to reveal silent constraint omission. Combined with differential testing, it forms a dual verifier. We instantiate and evaluate it on vehicle routing problems (VRPs), a representative constraint-dense combinatorial optimization testbed with coupled operational constraints. We develop VRPCoder, an 8B end-to-end model that translates natural-language VRP scenarios into Gurobi scripts, together with an expert-verified VRP benchmark suite covering 21 variants. The verifier is reused as a rejection-sampling filter during data synthesis and as a per-rollout reward in group relative policy optimization (GRPO). Across four VRP benchmarks, VRPCoder-GRPO reaches 93\\% average Pass@1, outperforms Gemini-3.1-Pro Preview on three benchmarks, exceeds Claude-Sonnet-4.5 by 28 average points, and surpasses prior OR-LLMs by 78 average points.","short_abstract":"Large language models (LLMs) increasingly translate natural-language optimization problems into executable solver code. Yet for constraint-dense operations research (OR) problems, existing data-filtering and training pipelines largely rely on objective-equivalence signals such as differential testing and answer agreeme...","url_abs":"https://arxiv.org/abs/2606.04816","url_pdf":"https://arxiv.org/pdf/2606.04816v1","authors":"[\"Xizi Luo\",\"Changhong He\",\"Dongdong Geng\",\"Chenggong Shi\",\"Yu Mei\"]","published":"2026-06-03T12:39:04Z","proceeding":"cs.AI","tasks":"[\"cs.AI\",\"cs.LG\"]","methods":"[\"Large Language Model\",\"Language Model\"]","has_code":false}