Three methods, one problem: Classical and AI approaches to no-three-in-line

TL;DR

This paper compares classical ILP methods with AI techniques like PatternBoost and reinforcement learning for the no-three-in-line problem, highlighting their strengths and limitations in solving larger grid instances.

Contribution

It provides the first systematic comparison of classical and AI approaches to the no-three-in-line problem, including applying transformer learning and reinforcement learning.

Findings

ILP achieves optimal solutions up to 19x19 grids.

PatternBoost reduces test loss by 96% up to 14x14 grids.

PPO finds perfect solutions on 10x10 grids but struggles at 11x11.

Abstract

The No-Three-In-Line problem asks for the maximum number of points that can be placed on an n by n grid with no three collinear, representing a famous problem in combinatorial geometry. While classical methods like Integer Linear Programming (ILP) guarantee optimal solutions, they face exponential scaling with grid size, and recent advances in machine learning offer promising alternatives for pattern-based approximation. This paper presents the first systematic comparison of classical optimization and AI approaches to this problem, evaluating their performance against traditional algorithms. We apply PatternBoost transformer learning and reinforcement learning (PPO) to this problem for the first time, comparing them against ILP. ILP achieves provably optimal solutions up to 19 by 19 grids, while PatternBoost matches optimal performance up to 14 by 14 grids with 96% test loss reduction. PPO achieves perfect solutions on 10 by 10 grids but fails at 11 by 11 grids, where constraint violations prevent valid configurations. These results demonstrate that classical optimization remains essential for exact solutions while AI methods offer competitive performance on smaller instances, with hybrid approaches presenting the most promising direction for scaling to larger problem sizes.