Geometry-Aware MCTS for Extremal Problems in Combinatorial Geometry

2026-06-24

AI Summary

This work addresses extremal point configuration problems in combinatorial geometry under strict global constraints such as the No-Three-in-Line (N3IL) condition, where traditional approaches suffer from combinatorial explosion and sparse rewards. The authors propose the first geometry-aware Monte Carlo Tree Search (MCTS) framework that directly embeds geometric constraints into the action space, enabling incremental O(n²) collinearity verification. By integrating symmetry-based pruning and batched state transitions, the method achieves substantial computational efficiency gains. It establishes new best-known results on five of six classical benchmarks: constructing Max-N3IL configurations of size up to 1.8n on grids with 82 ≤ n ≤ 119, and reducing the upper bound for the Smallest Complete Set problem to approximately 0.95n, thereby significantly enhancing scalability and solution capability.

Abstract

We study certain extremal problems in combinatorial geometry that ask about configurations of points in an $n \times n$ grid that satisfy strict, global geometric constraints. Classical exact solvers suffer from combinatorial explosion for these types of problems, and standard reinforcement learning and transformer-based models struggle with the sparse reward "validity cliff" and quadratic token-consumption limits. To overcome these bottlenecks, we propose a Geometry-Aware Monte Carlo Tree Search (MCTS) framework. Our approach strictly enforces geometric constraints through incremental updates to the feasible action space. For constraints about collections of collinear points, like those that occur in the classic No-Three-in-Line problem (Max-N3IL), this mechanism reduces the constraint checking complexity from $O(n^3)$ to $O(n^2)$. To improve search efficiency, we exploit geometric symmetries in two ways: canonical pruning during node expansion to reduce the branching factor, and symmetric batch transitions to accelerate the discovery of promising configurations. We perform extensive experiments and establish new best-known computational results on five out of six of the problems that we considered. Notably, for Max-N3IL we find configurations of size roughly $1.8 n$ for grids of size $82 \le n \le 119$. For the Smallest Complete Set problem, we find configurations of size roughly $0.95 n$, providing new upper bounds within the tested grids. This work establishes Geometry-Aware MCTS as a highly adaptable framework for discovering novel configurations in combinatorial geometry.

Problem

• combinatorial geometry
• extremal problems
• No-Three-in-Line
• geometric constraints
• point configurations

Innovation

• Geometry-Aware MCTS
• combinatorial geometry
• constraint enforcement
• symmetry exploitation
• extremal problems