RadonBench: A Benchmark of Classical and Learning-Guided Search Techniques for Problems in Combinatorial Geometry

Classical optimization and AI-based search offer complementary approaches to open problems in discrete mathematics, but a systematic comparison of these approaches on a common task is lacking. We benchmark five representative classical and AI-based methods on the integer Radon problem, which asks for large sets of lattice points that cannot be partitioned into two nonempty parts whose convex hulls share a lattice point. The Radon number $$\begin{gathered} r( \\ mathbb{Z}^d) \end{gathered}$$ is the smallest $n$ such that every $n$-point subset of $mathbb{Z}^d$ admits such a partition. The current bounds are $$\begin{gathered} 11 \\ ler( \\ mathbb{Z}^3) \\ le17 \end{gathered}$$ and $$\begin{gathered} 21 \\ ler( \\ mathbb{Z}^4) \\ le63 \end{gathered}$$. Finding a Radon-independent set of 11 points in $mathbb{Z}^3$ or 21 points in $mathbb{Z}^4$ would raise the corresponding lower bound to 12 or 22. We propose RadonBench, which includes exhaustive extension, greedy local search, and task-specific adaptations of PatternBoost, AlphaZero, and AlphaEvolve-style program evolution. The three AI approaches respectively learn point proposals, guide Monte Carlo tree search with a set-attention policy and value network, and evolve candidate-generating programs. To compare their different output types, we report target recovery, the largest valid construction, and the number of shared Radon points in target-sized near misses, where zero certifies a valid construction. PatternBoost and AlphaZero-Radon construct Radon-independent sets with 10 points in $mathbb{Z}^3$ and 20 points in $mathbb{Z}^4$, matching the current best-known sizes. At the open targets, AlphaZero-Radon and exhaustive extension reach a minimum of two shared Radon points, while greedy local search and AlphaEvolve-style program evolution reach one. None of the five methods finds a target-sized Radon-independent set. Thus, whether $$\begin{gathered} r( \\ mathbb{Z}^3) \\ ge12 \end{gathered}$$ or $$\begin{gathered} r( \\ mathbb{Z}^4) \\ ge22 \end{gathered}$$ remains open.