Can LLMs Prove Robotic Path Planning Optimality? A Benchmark for Research-Level Algorithm Verification

1. Problem Statement

Robotic path planning problems are typically NP-hard, necessitating the use of approximation algorithms with provable performance guarantees (approximation ratios). While designing these algorithms is challenging, formally proving their optimality is even more demanding. It requires:

• Domain-specific geometric insights: Understanding constraints like energy budgets, terrain, and visibility.
• Multi-step mathematical reasoning: Connecting concrete task descriptions to abstract mathematical conditions (e.g., mapping tour planning to combinatorial structures like MSTs and matchings).
• Long-context reasoning: Synthesizing algorithm descriptions, environmental constraints, and theoretical guarantees simultaneously.

The core research question is: Can Large Language Models (LLMs) assist in or automate the formal verification of approximation optimality for robotic path planning algorithms? This task is distinct from standard math benchmarks (like MATH or GSM8K) because it operates at a research level, involving complex, constrained, and domain-specific logic rather than self-contained algebraic problems.

2. Methodology

A. Benchmark Construction (RoboticBench)

The authors introduced the first benchmark specifically for evaluating LLMs on approximation-ratio proofs in robotics.

• Dataset: 34 research-grade proof tasks extracted from 11 peer-reviewed robotic papers.
• Task Types: The tasks cover diverse constrained planning problems, including:
  • Energy-constrained coverage path planning.
  • Distance/capacity-constrained vehicle routing (DVRP, CVRP).
  • Geometric neighborhood touring (e.g., Cone TSPN in 3D).
• Data Processing: An automated pipeline (using GPT-5.2) extracted problem definitions, algorithm descriptions, ground-truth proofs, and key lemmas from PDFs. Crucially, problem and algorithm names were anonymized to prevent LLMs from retrieving memorized proofs via data contamination.
• Evaluation Metrics: An LLM-as-Judge (GPT-5.2) evaluated generated proofs on a 1–10 scale across three dimensions:
  1. Final Answer: Accuracy of the predicted approximation ratio.
  2. Reasoning Correctness: Logical validity, absence of unjustified steps, and correct inequality directions.
  3. Proof Relevance: Grounding in the provided context (lemmas/definitions) vs. hallucination.

B. Experimental Settings

The authors evaluated state-of-the-art models (GPT-5.2, Gemini 3 Pro, Grok 4.1, Qwen3.5) under four information availability settings:

1. Setting 1 (No Context): Only problem description and algorithm.
2. Setting 2 (Context-Augmented): Problem + Algorithm + In-context Lemmas (key theorems extracted from the source paper).
3. Setting 3 (Posterior-Augmented): Problem + Algorithm + Ground-truth Approximation Ratio (but no lemmas).
4. Setting 4 (Full Augmentation): Problem + Algorithm + Lemmas + Ground-truth Ratio.

Additionally, they tested the impact of Chain-of-Thought (CoT) prompting.

3. Key Contributions

1. First Research-Level Benchmark: Established a benchmark for evaluating LLMs on the formal verification of robotic path planning algorithms, moving beyond simple instance solving to theoretical proof generation.
2. Granular Error Analysis: Developed a taxonomy for reasoning failures, distinguishing between Logical Errors (e.g., overgeneralization, missing edge cases, incorrect inequalities) and Hallucinations (e.g., unsupported assumptions, irrelevant reasoning).
3. Novel Evaluation Metric: Introduced the "First Error Position" metric, tracking where in the proof chain the first logical failure occurs to measure reasoning stability.
4. Open Pipeline: Released an automated pipeline for extracting and formalizing path planning algorithms from research papers.

4. Key Results

A. Performance Without Context

• Struggle with Pure Reasoning: Even the strongest models (GPT-5.2) failed to produce valid proofs in Setting 1 (no context). The success rate was only 26.47%, with most models scoring below 5/10 on reasoning correctness.
• Dominant Error Mode: Without domain knowledge, unsupported inference and oversimplification of problem constraints were the most frequent errors.

B. Impact of Augmentation

• In-Context Lemmas are Critical: Providing task-specific lemmas (Setting 2) yielded the most significant improvement.
  • Success rates increased (e.g., Qwen3.5 jumped from 8.82% to 44.12%).
  • Qwen3.5 (open-source) outperformed proprietary models in this setting, suggesting strong latent capacity when guided by domain knowledge.
• Posterior Knowledge (Oracle Ratio): Providing the ground-truth ratio (Setting 3) helped slightly but was far less effective than providing lemmas.
• Synergy: Setting 4 (Lemmas + Ratio) achieved the highest performance (up to 50% success rate for GPT-5.2), indicating that the ratio acts as a consistency check after the proof skeleton is grounded by lemmas.
• CoT Limitations: Generic Chain-of-Thought prompting alone did not significantly reduce task difficulty. It only provided marginal benefits when combined with posterior knowledge.

C. Error Analysis Insights

• Error Propagation: There is a strong negative correlation between the Average Reasoning Score and the First Error Position. Proofs that fail early tend to have low overall scores, as early errors compound into incorrect deductions.
• Mitigation: In-context lemmas not only reduced the frequency of errors but also delayed the first error, allowing models to sustain coherent reasoning for longer.
• Failure Patterns: Failures often stemmed from misinterpreting problem definitions (e.g., reducing constrained problems to classical TSP) or applying lemmas without verifying preconditions.

5. Significance and Future Directions

• Limitations of Current LLMs: The study demonstrates that current LLMs cannot autonomously generate research-level proofs for complex robotic algorithms without external domain scaffolding. They lack the specific geometric and operational intuition required for these tasks.
• The Role of Domain Knowledge: The results suggest that for high-stakes scientific reasoning, curated in-context knowledge (lemmas) is far more effective than prompting strategies (CoT) or answer keys (posterior ratios).
• Future Work:
  • Multimodal Integration: Using Vision-Language Models (VLMs) to visualize geometric structures (e.g., visibility cones, tours) to verify intermediate claims.
  • Knowledge Libraries: Building a standardized, retrievable library of domain lemmas and proof schemas.
  • Proof-Aware Training: Developing training objectives that penalize unsupported inferences and reward the correct citation of provided lemmas at specific reasoning steps.

In conclusion, while LLMs show promise as assistants in algorithm verification, they currently function best as reasoning engines that require heavy domain scaffolding rather than autonomous proof generators for research-level robotic path planning.