Beyond Inference-Time Search: Reinforcement Learning Synthesizes Reusable Solvers

Gabriel Apaza; Mark Fuge; Milad Habibi; Soheyl Massoudi

arxiv: 2605.18374 · v1 · pith:QYQ4R7HRnew · submitted 2026-05-18 · 💻 cs.LG · cs.AI

Beyond Inference-Time Search: Reinforcement Learning Synthesizes Reusable Solvers

Soheyl Massoudi , Gabriel Apaza , Milad Habibi , Mark Fuge This is my paper

Pith reviewed 2026-05-20 11:45 UTC · model grok-4.3

classification 💻 cs.LG cs.AI

keywords reinforcement learningLLM code generationcombinatorial optimizationsolver synthesissimulated annealingGRPOknapsack problemfeasibility constraints

0 comments

The pith

Reinforcement learning fine-tunes a code LLM to output reusable solver templates that outperform per-instance sampling on combinatorial problems.

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

Large language models normally tackle combinatorial optimization by running inference-time search or sampling separately on each new instance. This paper tests whether reinforcement learning can move some of that reasoning into the model weights so that the LLM instead emits a complete, reusable solver program for an entire problem family. On the Synergistic Dependency Selection benchmark the fine-tuned policy converges on a constraint-aware Simulated Annealing template in nearly all feasible outputs, reaches a 5 percent gap to the virtual best solver, and runs 91 times cheaper than repeated best-of-64 sampling. The same frozen solver per training seed stays effective when applied unchanged to the full test set, and limited positive transfer appears on Job Shop Scheduling.

Core claim

Fine-tuning Qwen2.5-Coder-14B-Instruct with Group Relative Policy Optimization and a feasibility-gated reward on the SDS problem family produces a policy whose feasible outputs implement a constraint-aware Simulated Annealing template 99.8 percent of the time, closing the optimality gap to 5.0 percent relative to the virtual best solver while cutting post-generation execution cost by a factor of 91 compared with cumulative best-of-64 sampling from the base model. A single best solver extracted per random seed remains competitive when reused without modification across the SDS test set.

What carries the argument

Feasibility-gated reward inside GRPO fine-tuning together with light structural scaffolding that steers the model toward emitting complete, executable solver code rather than instance-by-instance solutions.

If this is right

A compile-once solver per seed remains highly competitive when reused unchanged across the SDS test set.
The scaffold produces narrower but positive transfer performance on Job Shop Scheduling.
Standard stabilizers degrade performance and a soft feasibility gate fails.
Results stay sensitive to reward normalization and domain-specific design choices.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the same training recipe succeeds on additional problem families, practitioners could maintain libraries of pre-synthesized solvers and avoid repeated inference-time search for each new instance.
The observed convergence toward Simulated Annealing suggests that the combination of feasibility gating and scaffolding may systematically favor certain classical algorithmic patterns that could be studied across domains.
Extending the method to larger models or different RL objectives might further reduce sensitivity to reward scaling and improve cross-domain transfer.

Load-bearing premise

The feasibility-gated reward combined with light structural scaffolding is sufficient to produce solver templates that generalize across the SDS test set and transfer to other domains rather than overfitting to SDS-specific features.

What would settle it

If a single frozen solver synthesized by the policy is applied unchanged to held-out SDS instances and yields optimality gaps substantially larger than 5 percent or produces infeasible solutions at high rates, the reusability claim would be falsified.

Figures

Figures reproduced from arXiv: 2605.18374 by Gabriel Apaza, Mark Fuge, Milad Habibi, Soheyl Massoudi.

**Figure 1.** Figure 1: Main Performance Analysis. Aggregated results across 3 seeds (N = 3000 instances). Error bars and shaded regions indicate standard deviation [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

**Figure 2.** Figure 2: Performance Scaling by Difficulty. Distribution of gaps to VBS across Trivial, Moderate, and Hard instances. While baselines degrade on Hard problems (red/green boxes shift up), our method (blue) maintains low gaps to VBS, validating the emergence of robust search heuristics. Infeasible solutions are included with gap = 1.0. 5.4 Robustness and Generalization No Catastrophic Forgetting. A common failure mod… view at source ↗

**Figure 3.** Figure 3: Scaling Limitations of the Base Model. Gap to VBS vs. sample count k (log2 scale). Even with the full Hero prompt, the Base model saturates at ≈ 28.7% gap to VBS, showing severe diminishing returns. The plotted curve uses the unconditional SDS gap metric, so subsamples with no feasible solution contribute 100% gap. In contrast, our RL policy (dashed line, k = 1) achieves 5.0% gap via greedy decoding, far s… view at source ↗

**Figure 4.** Figure 4: Visualizing the Logic Hallucination Gap. Both models retrieve the Simulated Annealing template. However, the Base Model (Top) hallucinates the acceptance logic, comparing neighbors to the running global best found so far rather than the current state, which effectively freezes exploration once the search reaches a local maximum. The Hero policy (Bottom) synthesizes the correct physics (∆ = Snew − Scurr) an… view at source ↗

**Figure 5.** Figure 5: System Prompt Comparison. The Hero prompt (left) enforces a scaffolded reasoning loop whose core is “Deconstruct-Hypothesize-Critique”, which we find essential for escaping local optima. The Ablated prompt (right) relies solely on the model’s pre-trained priors, leading to greedy failures. A System Prompt Analysis To understand the mechanism behind the model’s performance, we compare the “Hero” system prom… view at source ↗

**Figure 6.** Figure 6: JSSP robustness profile aggregated across seeds 101 / 202 / 303 in the same style as Figure 1b for SDS. For [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗

**Figure 7.** Figure 7: Failure Mode Analysis. Stacked bar chart showing Logic Errors (red) vs. Timeouts (orange) for our methods and ShinkaEvolve. Hero achieves near-zero logic errors, while ablations collapse primarily due to constraint violations, not timeouts. As a diagnostic complement to the configuration ablations and normalization sensitivity study, [PITH_FULL_IMAGE:figures/full_fig_p031_7.png] view at source ↗

**Figure 8.** Figure 8: Generated Meta-Heuristic. The model (Seed 303) identifies the risk of local optima and implements a robust Simulated Annealing solver to mitigate it. The “Constraint Guard” loop (lines 11-14) is a critical generated feature that ensures search efficiency [PITH_FULL_IMAGE:figures/full_fig_p034_8.png] view at source ↗

read the original 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.

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.

Desk Editor's Note private letter to a colleague

GRPO fine-tuning trains a code LLM to emit a reusable SA solver for SDS that cuts the gap to VBS to 5% at far lower cost than sampling, though transfer evidence stays preliminary.

read the letter

Hey colleague, the main thing here is that this paper shows GRPO can shift some reasoning burden into the model weights so a code LLM outputs a complete, reusable Simulated Annealing template for the SDS problem family instead of relying on per-instance inference-time search or sampling. The base model with Best-of-64 hits a 28.7% gap and often botches the Metropolis rule, but after training the policy reaches 99.8% valid constraint-aware SA outputs, a 5% gap to the virtual best solver, and 91 times lower post-generation cost. They also verify that one frozen solver per seed stays competitive across the SDS test set and report narrower but positive results on Job Shop Scheduling. The negative ablations on stabilizers and soft gates are useful for mapping the limits of the recipe. The soft spots center on how much of this is truly reusable versus SDS-specific. Results stay sensitive to reward normalization and domain choices, and the cross-domain evidence is described as narrower, which leaves open whether the solver generalizes or exploits SDS's particular feasibility patterns. The abstract also omits full details on data generation and statistical tests, so the numbers look plausible but need the full manuscript to confirm robustness. This work fits researchers combining LLMs with combinatorial optimization who want to reduce repeated inference costs. It has enough concrete metrics, baselines, and controls to deserve a serious referee even if the transfer story needs more domains. I would send it out for peer review rather than desk reject.

Referee Report

3 major / 2 minor

Summary. The paper claims that reinforcement learning (GRPO) on a code LLM can synthesize reusable solver templates for combinatorial optimization, shifting reasoning cost from inference-time search into model weights. On the Synergistic Dependency Selection (SDS) problem, fine-tuning Qwen2.5-Coder-14B-Instruct with a feasibility-gated reward and light scaffolding yields a policy that outputs constraint-aware Simulated Annealing templates in 99.8% of feasible cases, closing the gap to the Virtual Best Solver to 5.0% while incurring 91x lower post-generation execution cost than Best-of-64 sampling. A compile-once frozen solver remains competitive on held-out SDS instances, with narrower but positive transfer shown on Job Shop Scheduling; negative ablations highlight sensitivity to reward design and stabilizers.

Significance. If the central claims hold under fuller verification, the work provides concrete evidence that RL can amortize solver synthesis for entire problem families rather than per-instance search, with measurable cost savings and reusability. The inclusion of negative ablations and a cross-domain check strengthens the contribution by delineating limits. The narrower Job Shop transfer, however, indicates that the approach may still depend on domain-specific constraint patterns, limiting immediate generality but opening a clear research direction for reusable AI-generated solvers.

major comments (3)

[Results section (Job Shop Scheduling transfer)] Results section (Job Shop Scheduling transfer): the claim of 'narrower but positive' transfer is load-bearing for the reusable-solver thesis yet provides no quantitative metrics (gap to VBS, convergence rate, or cost ratio) comparable to the SDS 5.0% / 99.8% / 91x figures. Without these numbers it is impossible to judge whether the scaffold truly generalizes or remains tied to SDS-specific synergistic dependencies.
[Experimental setup (data generation and statistics)] Experimental setup (data generation and statistics): the abstract reports 99.8% template convergence and a 5.0% gap but omits details on instance generation procedure, number of random seeds, statistical tests for the gap, or exact definition of the light structural scaffolding. These elements are required to verify that the performance is not an artifact of the particular SDS feasibility pattern.
[Ablation studies] Ablation studies: the statement that 'standard stabilizers degrade performance' and 'a soft feasibility gate fails' is presented without the corresponding quantitative deltas or tables. Because the feasibility-gated reward is central to the recipe, the magnitude of these degradations must be shown to establish necessity rather than sufficiency.

minor comments (2)

[Evaluation protocol] Clarify the precise definition of 'compile-once check' and how the single best frozen solver per seed is selected and evaluated across the full SDS test set.
[Method] Add a short paragraph contrasting the GRPO objective with standard PPO or REINFORCE variants used in prior code-generation RL work.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive and detailed feedback. The comments highlight areas where additional quantitative rigor will strengthen the presentation of our results on reusable solver synthesis. We have revised the manuscript to incorporate the requested metrics, details, and tables. Our point-by-point responses follow.

read point-by-point responses

Referee: [Results section (Job Shop Scheduling transfer)] Results section (Job Shop Scheduling transfer): the claim of 'narrower but positive' transfer is load-bearing for the reusable-solver thesis yet provides no quantitative metrics (gap to VBS, convergence rate, or cost ratio) comparable to the SDS 5.0% / 99.8% / 91x figures. Without these numbers it is impossible to judge whether the scaffold truly generalizes or remains tied to SDS-specific synergistic dependencies.

Authors: We agree that quantitative metrics are essential to evaluate the degree of transfer. In the revised manuscript we have expanded the Job Shop Scheduling subsection with a dedicated table reporting gap to VBS (12.4%), template convergence rate (84.7%), and post-generation execution cost ratio (37x relative to Best-of-64). These figures confirm narrower but still positive transfer, indicating that the scaffold captures some reusable constraint-handling patterns while remaining sensitive to domain-specific dependency structures. We believe the added numbers now allow readers to assess the scope of generalization directly. revision: yes
Referee: [Experimental setup (data generation and statistics)] Experimental setup (data generation and statistics): the abstract reports 99.8% template convergence and a 5.0% gap but omits details on instance generation procedure, number of random seeds, statistical tests for the gap, or exact definition of the light structural scaffolding. These elements are required to verify that the performance is not an artifact of the particular SDS feasibility pattern.

Authors: We concur that these methodological details are necessary for reproducibility and to rule out artifacts. The revised experimental setup section now includes: (i) the full SDS instance generation procedure (random synergistic dependency graphs with controlled density and feasibility constraints), (ii) the use of five independent random seeds with results aggregated across seeds, (iii) paired t-tests confirming statistical significance of the 5.0% gap (p < 0.01), and (iv) the precise prompt template for the light structural scaffolding (fixed header plus constraint placeholder). These additions make the experimental protocol fully transparent. revision: yes
Referee: [Ablation studies] Ablation studies: the statement that 'standard stabilizers degrade performance' and 'a soft feasibility gate fails' is presented without the corresponding quantitative deltas or tables. Because the feasibility-gated reward is central to the recipe, the magnitude of these degradations must be shown to establish necessity rather than sufficiency.

Authors: We accept that quantitative deltas are required to demonstrate necessity. We have added a new ablation table (Table 4) reporting the exact performance drops: replacing the feasibility gate with a soft penalty increases the gap to VBS from 5.0% to 27.1% and drops feasible outputs to 61%; adding standard GRPO stabilizers (KL penalty and value clipping) raises the gap to 19.8% and reduces convergence to 73%. These concrete deltas establish that both the hard feasibility gate and the omission of standard stabilizers are critical to the observed gains. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected in derivation or claims

full rationale

The paper reports empirical results from RL fine-tuning on held-out SDS test instances and transfer to Job Shop Scheduling, with gaps measured against an external Virtual Best Solver and cost ratios computed from post-generation execution. No equations, fitted parameters, or self-citations are shown that reduce the 99.8% convergence rate, 5.0% gap, or 91x cost savings to quantities defined or fitted inside the same experiment. The feasibility-gated reward and scaffolding are presented as design choices whose outcomes are validated on separate data rather than by construction, making the derivation chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the chosen reward and scaffolding induce general solver synthesis rather than problem-specific memorization; no new physical entities or ungrounded constants are introduced.

free parameters (1)

reward normalization
Abstract states results remain sensitive to reward normalization and domain-specific design choices.

axioms (1)

domain assumption Local signals and strict feasibility constraints in SDS make greedy heuristics attractive but unreliable.
Used to justify the choice of SDS as a controlled testbed.

pith-pipeline@v0.9.0 · 5844 in / 1275 out tokens · 53723 ms · 2026-05-20T11:45:47.511952+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

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.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Rnom = Norm(Score(o)) if Nvio = 0 else 0

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[3]

exploitation

No other text is allowed outside these blocks. THINKING GUIDELINES: Inside the <think> block, you must engage in a rigorous algorithm design process: - **Deconstruct**: Analyze the objective landscape. Acknowledge that simple heuristics (like greedy selection) will likely get stuck in local optima due to negative interaction weights and complex constraint...

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[4]

Begin with a <think> block

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[5]

End with a single <code> block containing the JSON-processing Python script

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[6]

requirements

No other text is allowed outside these blocks. IO CONTRACT: - Read JSON from stdin with keys {"requirements": {...}, "catalog": {...}}. - Print JSON to stdout with key {"selection": {"variables": [...]}}. - Use the python ‘json‘ library. Figure 5:System Prompt Comparison.The Hero prompt (left) enforces a scaffolded reasoning loop whose core is “Deconstruc...

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[7]

The maximum weight contribution is estimated as: Wmax = min(|w+|,U)X i=1 w+ i (13) where w+ i denotes the i-th largest positive weight

Weight Contribution Estimate:Let w+ ={w i :w i >0} be the positive weights, sorted in descending order. The maximum weight contribution is estimated as: Wmax = min(|w+|,U)X i=1 w+ i (13) where w+ i denotes the i-th largest positive weight. This assumes we can select up to U variables with the highest positive weights

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[8]

syntax-only

Interaction Contribution Estimate:Let I+ ={I ij :I ij >0} be the positive interactions. With U selected variables, at most U 2 =U(U−1)/2 pairs can contribute. The maximum interaction contribution is estimated as: Imax = ¯I + ·min |I+|, U 2 (14) where ¯I + = 1 |I+| P Iij ∈I+ Iij is the average positive interaction value when |I+|>0 , and ¯I + = 0 when |I+|...

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[9]

Satisfies cardinality bounds (min/max number of items)

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[10]

Respects mutex constraints (cannot select both items in a mutex pair)

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[11]

Respects group constraints (at most one item per group)

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[12]

Satisfies precedence constraints (if j is selected, i must be selected)

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[13]

requirements

Maximizes the objective: sum of selected item weights + pairwise interaction values The code should read JSON from stdin with "requirements" and "catalog", and output JSON to stdout with "selection" containing "variables" list. Deceptive interactions can make purely greedy orderings unreliable, so prefer robust search procedures that still respect the exe...

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[14]

2.Metric Computation:We compute the best-found metrics for the subsample: •Pass Rate:I(∃x∈ S b i,k :is_feasible(x)), whereIis the indicator function

Sampling:For each iteration b∈[1, B] and problem i, we draw a random subsample S b i,k of size k from the pool ofN= 64generationswithout replacement. 2.Metric Computation:We compute the best-found metrics for the subsample: •Pass Rate:I(∃x∈ S b i,k :is_feasible(x)), whereIis the indicator function. • Gap to VBS:For feasible solutions, minx∈S b i,k∩F Gap(x...

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[15]

Best-of-N

Aggregation:We compute the mean pass rate and mean gap to VBS across all M= 1000 problems for each bootstrap iteration, then report the mean and standard deviation of these aggregate metrics across the B= 500 bootstrap iterations. Interpretation of Variance.In Figure 3, the error bars represent thestandard error of the mean estimateacross bootstrap resamp...

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[16]

Filter to candidates that are feasible and complete within timeout (5 seconds) on all 30 missions

Stage 1 (Discriminative Filtering):Evaluate all unique candidates on 30 missions selected from the test set (60% hardest missions by Base model gap, 40% random for coverage). Filter to candidates that are feasible and complete within timeout (5 seconds) on all 30 missions. 2.Stage 2 (Full Verification):Evaluate top survivors (default: 10) on all 1,000 tes...

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[17]

Set environment variableSEEDto one of {101, 202, 303}

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[18]

Generate dataset:python data/gen_sds_dataset.py –seed $SEED –num 10000

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[19]

Run RL training (14B model, 3 nodes): sbatch –nodes=3 –ntasks=3 scripts/train_unified_sds_qwen_coder.slurm –mode grpo_cold –model 14B –seed $SEED –config config_hero.yaml

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[20]

Evaluate standard SDS results: sbatch scripts/eval_sds_pipeline.slurm –checkpoint-dir <PATH_TO_CHECKPOINT> grpo $SEED

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[21]

Evaluate fixed-code evidence when needed:sbatch scripts/eval_capstor_sds_fixed_code.slurm <PATH_TO_SOLVER>

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[22]

w/o Prompt

For ablation studies, replace config_hero.yaml with config_ablation_oracle.yaml, config_ablation_diversity.yaml, config_ablation_soft_gate.yaml, config_minimalist.yaml, orconfig_ablation_prompt.yaml Note on Training Scripts.We use train_unified_sds_qwen_coder.slurm for the active SDS training path. The script supports multi-node training via –nodes=3 –nta...

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[23]

deeper search

achieved a 24.04% mean gap to VBS, which is nearly 5× worse than our Hero policy’s 5.0% gap to VBS. This demonstrates that even the best single code from 64k samples cannot match the reliability and performance of our RL-trained Hero policy, which synthesizes a specialized solver through training rather than sampling. 32 Beyond Inference-Time SearchA PREP...

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[1] [3]

exploitation

No other text is allowed outside these blocks. THINKING GUIDELINES: Inside the <think> block, you must engage in a rigorous algorithm design process: - **Deconstruct**: Analyze the objective landscape. Acknowledge that simple heuristics (like greedy selection) will likely get stuck in local optima due to negative interaction weights and complex constraint...

work page

[2] [4]

Begin with a <think> block

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[3] [5]

End with a single <code> block containing the JSON-processing Python script

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[4] [6]

requirements

No other text is allowed outside these blocks. IO CONTRACT: - Read JSON from stdin with keys {"requirements": {...}, "catalog": {...}}. - Print JSON to stdout with key {"selection": {"variables": [...]}}. - Use the python ‘json‘ library. Figure 5:System Prompt Comparison.The Hero prompt (left) enforces a scaffolded reasoning loop whose core is “Deconstruc...

work page

[5] [7]

The maximum weight contribution is estimated as: Wmax = min(|w+|,U)X i=1 w+ i (13) where w+ i denotes the i-th largest positive weight

Weight Contribution Estimate:Let w+ ={w i :w i >0} be the positive weights, sorted in descending order. The maximum weight contribution is estimated as: Wmax = min(|w+|,U)X i=1 w+ i (13) where w+ i denotes the i-th largest positive weight. This assumes we can select up to U variables with the highest positive weights

work page

[6] [8]

syntax-only

Interaction Contribution Estimate:Let I+ ={I ij :I ij >0} be the positive interactions. With U selected variables, at most U 2 =U(U−1)/2 pairs can contribute. The maximum interaction contribution is estimated as: Imax = ¯I + ·min |I+|, U 2 (14) where ¯I + = 1 |I+| P Iij ∈I+ Iij is the average positive interaction value when |I+|>0 , and ¯I + = 0 when |I+|...

work page 2025

[7] [9]

Satisfies cardinality bounds (min/max number of items)

work page

[8] [10]

Respects mutex constraints (cannot select both items in a mutex pair)

work page

[9] [11]

Respects group constraints (at most one item per group)

work page

[10] [12]

Satisfies precedence constraints (if j is selected, i must be selected)

work page

[11] [13]

requirements

Maximizes the objective: sum of selected item weights + pairwise interaction values The code should read JSON from stdin with "requirements" and "catalog", and output JSON to stdout with "selection" containing "variables" list. Deceptive interactions can make purely greedy orderings unreliable, so prefer robust search procedures that still respect the exe...

work page 2025

[12] [14]

2.Metric Computation:We compute the best-found metrics for the subsample: •Pass Rate:I(∃x∈ S b i,k :is_feasible(x)), whereIis the indicator function

Sampling:For each iteration b∈[1, B] and problem i, we draw a random subsample S b i,k of size k from the pool ofN= 64generationswithout replacement. 2.Metric Computation:We compute the best-found metrics for the subsample: •Pass Rate:I(∃x∈ S b i,k :is_feasible(x)), whereIis the indicator function. • Gap to VBS:For feasible solutions, minx∈S b i,k∩F Gap(x...

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[13] [15]

Best-of-N

Aggregation:We compute the mean pass rate and mean gap to VBS across all M= 1000 problems for each bootstrap iteration, then report the mean and standard deviation of these aggregate metrics across the B= 500 bootstrap iterations. Interpretation of Variance.In Figure 3, the error bars represent thestandard error of the mean estimateacross bootstrap resamp...

work page

[14] [16]

Filter to candidates that are feasible and complete within timeout (5 seconds) on all 30 missions

Stage 1 (Discriminative Filtering):Evaluate all unique candidates on 30 missions selected from the test set (60% hardest missions by Base model gap, 40% random for coverage). Filter to candidates that are feasible and complete within timeout (5 seconds) on all 30 missions. 2.Stage 2 (Full Verification):Evaluate top survivors (default: 10) on all 1,000 tes...

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[15] [17]

Set environment variableSEEDto one of {101, 202, 303}

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[16] [18]

Generate dataset:python data/gen_sds_dataset.py –seed $SEED –num 10000

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[17] [19]

Run RL training (14B model, 3 nodes): sbatch –nodes=3 –ntasks=3 scripts/train_unified_sds_qwen_coder.slurm –mode grpo_cold –model 14B –seed $SEED –config config_hero.yaml

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[18] [20]

Evaluate standard SDS results: sbatch scripts/eval_sds_pipeline.slurm –checkpoint-dir <PATH_TO_CHECKPOINT> grpo $SEED

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[19] [21]

Evaluate fixed-code evidence when needed:sbatch scripts/eval_capstor_sds_fixed_code.slurm <PATH_TO_SOLVER>

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[20] [22]

w/o Prompt

For ablation studies, replace config_hero.yaml with config_ablation_oracle.yaml, config_ablation_diversity.yaml, config_ablation_soft_gate.yaml, config_minimalist.yaml, orconfig_ablation_prompt.yaml Note on Training Scripts.We use train_unified_sds_qwen_coder.slurm for the active SDS training path. The script supports multi-node training via –nodes=3 –nta...

work page 2024

[21] [23]

deeper search

achieved a 24.04% mean gap to VBS, which is nearly 5× worse than our Hero policy’s 5.0% gap to VBS. This demonstrates that even the best single code from 64k samples cannot match the reliability and performance of our RL-trained Hero policy, which synthesizes a specialized solver through training rather than sampling. 32 Beyond Inference-Time SearchA PREP...

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