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Towards Solving the Gilbert-Pollak Conjecture via Large Language Models

Yisi Ke, Tianyu Huang, Yankai Shu, Di He, Jingchu Gai, Liwei Wang

TL;DR

This work tackles the Gilbert–Pollak Steiner ratio conjecture by building an AI-driven, lemma-based discovery system. Rather than solving the conjecture head-on, the method constructs rule-constrained geometric lemmas that are implemented as executable verification functions, enabling a rigorous lower-bound computation via a minimax framework. Through iterative refinement with a bottleneck-reflection mechanism, the system achieves a new certified lower bound of $\rho_{Steiner}=0.8559$, improving the long-standing $0.824$ and narrowing the gap toward $\sqrt{3}/2$. The approach demonstrates that Large Language Models can contribute to high-level mathematical research when combined with formal verification and structured reasoning, with results backed by multiple backbones, reproducible code, and a concise computational budget. This suggests a scalable paradigm for AI-assisted mathematical discovery across combinatorial geometry and beyond.

Abstract

The Gilbert-Pollak Conjecture \citep{gilbert1968steiner}, also known as the Steiner Ratio Conjecture, states that for any finite point set in the Euclidean plane, the Steiner minimum tree has length at least $\sqrt{3}/2 \approx 0.866$ times that of the Euclidean minimum spanning tree (the Steiner ratio). A sequence of improvements through the 1980s culminated in a lower bound of $0.824$, with no substantial progress reported over the past three decades. Recent advances in LLMs have demonstrated strong performance on contest-level mathematical problems, yet their potential for addressing open, research-level questions remains largely unexplored. In this work, we present a novel AI system for obtaining tighter lower bounds on the Steiner ratio. Rather than directly prompting LLMs to solve the conjecture, we task them with generating rule-constrained geometric lemmas implemented as executable code. These lemmas are then used to construct a collection of specialized functions, which we call verification functions, that yield theoretically certified lower bounds of the Steiner ratio. Through progressive lemma refinement driven by reflection, the system establishes a new certified lower bound of 0.8559 for the Steiner ratio. The entire research effort involves only thousands of LLM calls, demonstrating the strong potential of LLM-based systems for advanced mathematical research.

Towards Solving the Gilbert-Pollak Conjecture via Large Language Models

TL;DR

This work tackles the Gilbert–Pollak Steiner ratio conjecture by building an AI-driven, lemma-based discovery system. Rather than solving the conjecture head-on, the method constructs rule-constrained geometric lemmas that are implemented as executable verification functions, enabling a rigorous lower-bound computation via a minimax framework. Through iterative refinement with a bottleneck-reflection mechanism, the system achieves a new certified lower bound of , improving the long-standing and narrowing the gap toward . The approach demonstrates that Large Language Models can contribute to high-level mathematical research when combined with formal verification and structured reasoning, with results backed by multiple backbones, reproducible code, and a concise computational budget. This suggests a scalable paradigm for AI-assisted mathematical discovery across combinatorial geometry and beyond.

Abstract

The Gilbert-Pollak Conjecture \citep{gilbert1968steiner}, also known as the Steiner Ratio Conjecture, states that for any finite point set in the Euclidean plane, the Steiner minimum tree has length at least times that of the Euclidean minimum spanning tree (the Steiner ratio). A sequence of improvements through the 1980s culminated in a lower bound of , with no substantial progress reported over the past three decades. Recent advances in LLMs have demonstrated strong performance on contest-level mathematical problems, yet their potential for addressing open, research-level questions remains largely unexplored. In this work, we present a novel AI system for obtaining tighter lower bounds on the Steiner ratio. Rather than directly prompting LLMs to solve the conjecture, we task them with generating rule-constrained geometric lemmas implemented as executable code. These lemmas are then used to construct a collection of specialized functions, which we call verification functions, that yield theoretically certified lower bounds of the Steiner ratio. Through progressive lemma refinement driven by reflection, the system establishes a new certified lower bound of 0.8559 for the Steiner ratio. The entire research effort involves only thousands of LLM calls, demonstrating the strong potential of LLM-based systems for advanced mathematical research.
Paper Structure (47 sections, 29 theorems, 50 equations, 11 figures, 1 table, 3 algorithms)

This paper contains 47 sections, 29 theorems, 50 equations, 11 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Preliminary
  4. Gilbert-Pollak Conjecture and Steiner Tree
  5. Mathematical Framework for Proving Steiner Ratio Lower Bound
  6. LLM Math System for the Conjecture
  7. Reward Model
  8. LLM Agent
  9. Finding Trapped Regular Point Lemma
  10. Finding Valid 4-Point Steiner Tree Lemma
  11. Iterative Process
  12. Experiments and Results
  13. Conclusion
  14. Appendix
  15. Mathematical Preliminaries
  16. ...and 32 more sections

Key Result

Theorem 4

Let $\mathcal{W}$ denote the space of all edge-length vectors associated with the tree topology. For any choice of splitting $\tau = (V^*, S^-, S^+, t^*)$ and geometric configuration ${\bm{w}} \in \mathcal{W}$, we define the splitting function$F_{\tau}({\bm{w}}, \rho)$ as: Let $\mathcal{F}$ be a family of functions corresponding to admissible splits $\tau$, which we denote as spliting function. T

Figures (11)

  • Figure 1: Progress on Steiner Ratio Lower Bound
  • Figure 2: Illustration of our proposed LLM Math Research System for the Gilbert-Pollak Conjecture
  • Figure 3: Illustration of minimum spanning tree (left) and Steiner minimal tree (right).
  • Figure 4: Illustration of Pruning Process. We try to prune $V^* = \{B\}$. Black dashed lines are $S^-$. Black solid lines are $S^+$. Green line is $t^*$.
  • Figure 5: Branch-and-Bound on 2D Plane
  • ...and 6 more figures

Theorems & Definitions (54)

  • Definition 1: Minimum Spanning Tree
  • Definition 2: Steiner Minimal Tree
  • Definition 3: Steiner Ratio
  • Theorem 4
  • Definition 5
  • Theorem 6
  • Lemma 7: Type A: Intrinsic Properties
  • Lemma 8: Type B: Sufficient Trapping Conditions
  • Theorem 9
  • Definition 10: Steiner Tree Type
  • ...and 44 more