Formalize, Don't Optimize: The Heuristic Trap in LLM-Generated Combinatorial Solvers

Many optimization problems arise repeatedly from a fixed but unknown distribution. Even when the worst-case problem is hard, this distribution may carry reusable structure, such as recurring geometry, decompositions, or resource patterns. We study how to infer such structure from sample instances and compile it into solver code that runs faster on future instances while preserving solution quality. Our central abstraction is a \emph{solver hint}: distribution-specific structure inferred from samples and used to specialize a solver. We prove that the empirically fastest sample-consistent solver generalizes in both correctness and runtime over fixed solver libraries, and that identifiable hints can be recovered from polynomially many samples. We instantiate the framework with LLM code agents on $21$ combinatorial-optimization distributions across $7$ problem classes. The synthesized solvers reach mean normalized quality $0.971$ while running orders of magnitude faster than classical heuristics, Gurobi, and time-limited exact backends, though they do not dominate every baseline on every family. Against LLM synthesis baselines, they are faster than one-shot Codex, one-shot Claude Code, and a best-of-$5$ open-model variant; they improve quality over Claude Code and best-of-$5$, while nearly matching Codex quality and running substantially faster. This isolates the contribution of the iterative synthesis loop without claiming uniform domination over every LLM baseline. On the PACE 2025 Dominating Set private instances, the synthesized solver is valid on all $100$ graphs and runs roughly $75\times$--$125\times$ faster than released competition solvers, within a few percent of their solution size. These results suggest LLM agents can discover distribution-specific computational shortcuts and compile them into efficient solver code.

Large language models (LLMs) typically approach combinatorial optimization as an inference-time procedure, solving each instance separately through sampling, search, or repeated prompting. We ask whether reinforcement learning can instead shift part of this reasoning cost into the weights of a code LLM, so that the model synthesizes a reusable solver for an entire problem family. We study this question on Synergistic Dependency Selection (SDS), a controlled variant of constrained Quadratic Knapsack designed to expose a specific failure mode: local signals and strict feasibility constraints make greedy heuristics attractive but unreliable. Under identical scaffolding, Best-of-64 base-model sampling saturates at an approximately 28.7% gap to the global Virtual Best Solver (VBS); code audits show that the base model often retrieves Simulated Annealing templates but misimplements the Metropolis acceptance rule. We fine-tune Qwen2.5-Coder-14B-Instruct with Group Relative Policy Optimization (GRPO) using a feasibility-gated reward and light structural scaffolding. The resulting policy converges to a constraint-aware Simulated Annealing template in 99.8% of feasible SDS outputs, achieves a 5.0% gap to that VBS, and is 91 times cheaper in post-generation execution/search cost than cumulative Best-of-64 evaluation. A compile-once check shows that one best frozen solver per seed remains highly competitive when reused unchanged across the SDS test set, while an additional-domain evaluation on Job Shop Scheduling provides narrower but positive evidence that the scaffold transfers beyond SDS. Negative ablations reveal the limits of this recipe: standard stabilizers degrade performance, a soft feasibility gate fails, and results remain sensitive to reward normalization and domain-specific design choices.

LLMs can solve program synthesis tasks but remain inefficient and unreliable on hard instances requiring large combinatorial search. Given a small set of reasoning traces, we use coding agents to compile them into reusable symbolic program synthesizers over constrained DSLs. The resulting solvers require no LLM calls at test time and are strong standalone systems: symbolic solver ensembles reach 91.3% accuracy on PBEBench-Lite and 84.7% on PBEBench-Hard, outperforming LLMs with test-time scaling for the latter by +16.3 percentage points at zero LLM inference cost. They also complement LLM search, improving PBEBench-Hard accuracy from 68.4% to 85.8% while reducing reported token usage by 78%, and raising SLR-Bench hard-tier accuracy from 34.4% to 58.0% in a neuro-symbolic hybrid setting. Compared to directly using coding agents as per-instance solvers, induced solvers are substantially more Pareto-efficient, amortizing a small one-time construction cost over many zero-token executions. Finally, most solvers transfer zero-shot to a real historical linguistics task - predicting sound changes in natural language data - reaching 80.1% accuracy under ensembling and recovering some plausible linguistic rules. Together, these results show that reasoning traces can be compiled into reusable symbolic solvers that solve many tasks directly, complement LLM inference on hard cases, and provide a scalable route to domain-general solver induction. We release code and data for reproducibility.