Beyond Inference-Time Search: Reinforcement Learning Synthesizes Reusable Solvers

TL;DR

This paper demonstrates that reinforcement learning can synthesize reusable solvers for combinatorial optimization problems, reducing inference costs and improving efficiency over traditional sampling methods.

Contribution

It introduces a reinforcement learning approach to generate reusable solvers for problem families, outperforming baseline heuristics and sampling methods in efficiency and accuracy.

Findings

RL-synthesized solvers achieve a 5.0% gap to the Virtual Best Solver.

The learned solver is 91 times cheaper in execution cost than baseline sampling.

The approach transfers to Job Shop Scheduling beyond the original problem domain.

Abstract

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.