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arxiv: 2605.08905 · v1 · submitted 2026-05-09 · 💻 cs.AI

Recognition: 2 theorem links

· Lean Theorem

Forge: Quality-Aware Reinforcement Learning for NP-Hard Optimization in LLMs

Haodong Duan, Kai Chen, Linyang Li, Qingwen Liu, Shengyuan Ding, Xiaozhe Li, Xinyu Fang, Yang Li

Authors on Pith no claims yet

Pith reviewed 2026-05-12 02:36 UTC · model grok-4.3

classification 💻 cs.AI
keywords LLMsreinforcement learningNP-hard optimizationquality-aware rewardsOPT-BENCHsuccess ratequality ratiogeneralization
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The pith

Quality-aware reinforcement learning on OPT-BENCH enables small LLMs to find high-quality solutions to NP-hard optimization problems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes OPT-BENCH as a framework for training and evaluating LLMs on NP-hard optimization tasks by extending RLVR with quality signals. It provides instance generators, quality verifiers, and continuous rewards across ten tasks so models can improve solution quality beyond mere feasibility. A 7B model trained on 15K examples reaches 93.1 percent success rate and 46.6 percent quality ratio, beating GPT-4o. The same training produces gains on mathematics, logic, knowledge, and instruction-following tasks. Quality-aware rewards drive a 28.8 percent improvement over binary feedback, and task diversity matters more than raw data volume for generalization.

Core claim

OPT-BENCH supplies scalable training infrastructure, a 1,000-instance benchmark measuring both Success Rate and Quality Ratio, and quality-aware rewards that replace binary correctness signals; training Qwen2.5-7B-Instruct-1M on 15K examples produces 93.1 percent SR and 46.6 percent QR while outperforming GPT-4o and transferring to unrelated reasoning domains.

What carries the argument

Quality-aware rewards that assign continuous scores reflecting solution optimality instead of binary feasibility checks.

Load-bearing premise

Quality verifiers can accurately score how close any solution is to the true optimum even when the optimum itself cannot be computed for the test instances.

What would settle it

Compute exact optimal solutions for a fresh set of NP-hard instances and measure whether the quality ratios produced by models trained on OPT-BENCH match the 46.6 percent benchmark figure.

Figures

Figures reproduced from arXiv: 2605.08905 by Haodong Duan, Kai Chen, Linyang Li, Qingwen Liu, Shengyuan Ding, Xiaozhe Li, Xinyu Fang, Yang Li.

Figure 1
Figure 1. Figure 1: Overview of the FORGE-ENGINE. The FORGE-BENCH encompasses 10 NP-hard optimization tasks across five categories (e.g., subset selection, path planning), designed to assess reasoning capabilities. An automated pipeline consisting of a Data Generator, Solution Validator, and Heuristic Solver ensures controllable data synthesis, rigorous evaluation, and scalable training. A case study on the Hamiltonian Circui… view at source ↗
Figure 2
Figure 2. Figure 2: FORGE-RLVR training pipeline with quality-aware RLVR. The model generates solutions with step-by￾step reasoning, which are evaluated through three components: (i) format verification checking output structure, (ii) feasibility verification ensuring constraint satisfaction, and (iii) quality assessment measuring optimality relative to heuristic baselines. The combined reward signal guides model optimization… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of RL training strategies during multi-task training, with performance evaluated on both [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Comparison between single linear curriculum learning and curriculum replay strategy under GRPO [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗
read the original abstract

Large Language Models (LLMs) have achieved remarkable success on reasoning benchmarks through Reinforcement Learning with Verifiable Rewards (RLVR), excelling at tasks such as math, coding, logic, and puzzles. However, existing benchmarks evaluate only correctness, while overlooking optimality, namely the ability to find the best solutions under constraints. We propose OPT-BENCH, the first comprehensive framework for training and evaluating LLMs on NP-hard optimization problems through quality-aware RLVR. OPT-BENCH provides three key components: a scalable training infrastructure with instance generators, quality verifiers, and optimal baselines across 10 tasks; a rigorous benchmark with 1,000 instances evaluating both feasibility, measured by Success Rate, and quality, measured by Quality Ratio; and quality-aware rewards that enable continuous improvement beyond binary correctness. Training on Qwen2.5-7B-Instruct-1M with 15K examples achieves 93.1% SR and 46.6% QR, significantly outperforming GPT-4o, which achieves 29.6% SR and 14.6% QR. Beyond optimization, training on OPT-BENCH transfers to diverse tasks, including mathematics (+2.2%), logic (+1.2%), knowledge (+4.1%), and instruction following (+6.1%). Our analysis reveals that quality-aware rewards improve solutions by 28.8% over binary rewards, and that task diversity drives generalization more than data quantity, offering insights into RLVR scaling for complex reasoning.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript introduces OPT-BENCH, a framework for training and evaluating LLMs on NP-hard optimization problems via quality-aware Reinforcement Learning with Verifiable Rewards (RLVR). It comprises instance generators, quality verifiers, and optimal baselines for 10 tasks; a 1,000-instance benchmark assessing feasibility via Success Rate (SR) and optimality via Quality Ratio (QR); and quality-aware rewards that support continuous improvement beyond binary correctness. Training Qwen2.5-7B-Instruct-1M on 15K examples yields 93.1% SR and 46.6% QR, outperforming GPT-4o (29.6% SR, 14.6% QR). Quality-aware rewards improve solutions by 28.8% over binary rewards, with positive transfer to mathematics (+2.2%), logic (+1.2%), knowledge (+4.1%), and instruction following (+6.1%). Task diversity is shown to drive generalization more than data quantity.

Significance. If the quality verifiers are shown to track true optimality, the work would be significant for extending RLVR to optimization quality on NP-hard problems rather than binary correctness alone. The release of generators, verifiers, and baselines for 10 tasks provides a reusable infrastructure that could accelerate research in this area. The transfer gains and analysis of diversity versus quantity offer concrete insights for scaling RLVR on complex reasoning tasks. The benchmark itself is a clear contribution even if the numerical claims require further substantiation.

major comments (2)
  1. [Abstract and §3 (Methods)] Abstract and §3 (Methods): The central numerical claims (93.1% SR / 46.6% QR, 28.8% gain from quality-aware rewards, and superiority over GPT-4o) rest on the Quality Ratio metric. For NP-hard problems where exact optima are unavailable on the 1,000-instance test set, the manuscript must demonstrate that the quality verifiers correlate with true optimality. Validation against known optima on smaller, solvable instances (via exhaustive search or exact solvers) is required; without it, QR and the reported gains risk reflecting verifier heuristics rather than genuine optimization improvement. This is load-bearing for the primary results.
  2. [§4 (Results) and Table 1 (presumed)] §4 (Results) and Table 1 (presumed): No error bars, standard deviations, or statistical tests accompany the reported SR and QR values or the 28.8% improvement. Given the stochasticity of LLM sampling and RL training, multiple independent runs are needed to establish that the outperformance over GPT-4o and the reward-type ablation are reliable rather than run-specific.
minor comments (3)
  1. [Title and Abstract] The title refers to 'Forge' while the abstract and body introduce 'OPT-BENCH'; clarify whether Forge is the RL method, the full system, or a synonym, and ensure consistent nomenclature throughout.
  2. [§4 (Results)] Missing details on baseline construction: how GPT-4o and other models were prompted or sampled for the benchmark (e.g., temperature, few-shot examples, decoding strategy) should be specified to enable reproduction.
  3. [§5 (Transfer Experiments)] The transfer results (+2.2% math, etc.) are reported without specifying the evaluation benchmarks or whether the same quality-aware training was used; add precise cross-task evaluation protocols.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help strengthen the manuscript. We provide point-by-point responses to the major comments and indicate the revisions we plan to make.

read point-by-point responses

  1. Referee: [Abstract and §3 (Methods)] Abstract and §3 (Methods): The central numerical claims (93.1% SR / 46.6% QR, 28.8% gain from quality-aware rewards, and superiority over GPT-4o) rest on the Quality Ratio metric. For NP-hard problems where exact optima are unavailable on the 1,000-instance test set, the manuscript must demonstrate that the quality verifiers correlate with true optimality. Validation against known optima on smaller, solvable instances (via exhaustive search or exact solvers) is required; without it, QR and the reported gains risk reflecting verifier heuristics rather than genuine optimization improvement. This is load-bearing for the primary results.

    Authors: We agree that validating the quality verifiers' correlation with true optimality is essential to substantiate the Quality Ratio metric and the reported improvements. The original manuscript describes the quality verifiers and optimal baselines but does not include explicit validation experiments on small instances. In the revised manuscript, we will add a validation study in Section 3, where we generate small instances solvable by exact methods (e.g., dynamic programming or solvers for subsets of tasks), compute true optima, and report correlation coefficients (such as Spearman rank correlation) between verifier scores and true quality. This will confirm that QR reflects genuine optimization progress. revision: yes

  2. Referee: [§4 (Results) and Table 1 (presumed)] §4 (Results) and Table 1 (presumed): No error bars, standard deviations, or statistical tests accompany the reported SR and QR values or the 28.8% improvement. Given the stochasticity of LLM sampling and RL training, multiple independent runs are needed to establish that the outperformance over GPT-4o and the reward-type ablation are reliable rather than run-specific.

    Authors: We recognize the need for statistical rigor in reporting results from stochastic processes like LLM sampling and RL training. The presented results are from single training runs, which limits the assessment of variability. To address this, we will conduct multiple independent runs with different random seeds for the main experiments in the revised version. We will update Table 1 and the results section to include mean values with standard deviations or error bars, and perform statistical tests (e.g., t-tests) to assess the significance of the differences over GPT-4o and between reward types. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical results rest on external benchmarks and verifiers rather than self-referential definitions or fitted predictions

full rationale

The paper presents an empirical RLVR training pipeline on OPT-BENCH, reporting measured Success Rate and Quality Ratio on a held-out 1,000-instance benchmark after training on 15K generated examples. No equations, derivations, or first-principles claims appear in the provided text that reduce the reported performance numbers (93.1% SR, 46.6% QR, 28.8% improvement) to quantities defined from the same fitted outputs or self-citations. The quality verifiers and optimal baselines are described as external components of the benchmark infrastructure; the superiority claims are statistical comparisons against GPT-4o on fixed test instances, not algebraic identities or renamings of inputs. This is the normal case of a self-contained empirical study whose central numbers are falsifiable against the benchmark rather than forced by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms, or invented entities; the framework extends standard RLVR with quality scoring but does not introduce new mathematical objects or fitted constants beyond the usual RL hyperparameters.

pith-pipeline@v0.9.0 · 5594 in / 1340 out tokens · 37927 ms · 2026-05-12T02:36:01.520363+00:00 · methodology

discussion (0)

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