arXiv subjects

Yankai Shu

Publications and source records attributed to Yankai Shu.

2 recordsLinked to original sources

arXiv2026

Towards Solving the Gilbert-Pollak Conjecture via Large Language Models

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.

cs.DM↗

arXiv2025

Investigating Advanced Reasoning of Large Language Models via Black-Box Environment Interaction

Existing tasks fall short in evaluating reasoning ability of Large Language Models (LLMs) in an interactive, unknown environment. This deficiency leads to the isolated assessment of deductive, inductive, and abductive reasoning, neglecting the integrated reasoning process that is indispensable for human-like discovery learning. We introduce a novel evaluation paradigm, \textit{black-box environment interaction}, to tackle this challenge. A black-box environment is defined by a hidden function that maps a specific set of inputs to outputs. LLMs are required to unravel the hidden function behind the black-box environment by interacting with it in given exploration turns, and reasoning over observed input-output pairs. Leveraging this idea, we build the \textsc{Oracle} benchmark which comprises 6 types of black-box task with 96 black-box environments. 19 modern LLMs are benchmarked. o3, a leading LLM from OpenAI, ranks first in 5 of the 6 tasks, achieving over 70\% accuracy on most easy black-box environments. But it still struggles with some hard black-box tasks, where the average performance drops below 40\%. Further analysis reveals a universal difficulty among LLMs: They lack the high-level planning capability to develop efficient and adaptive exploration strategies for hypothesis refinement. Code is available in https://github.com/lemonsis/Oracle_Benchmark.

cs.AI↗