North-East Lattice Paths Avoiding $k$ Collinear Points via Satisfiability

2025-11-28

AI Summary

This study addresses the extremal problem of determining the maximum length of a northeast lattice path avoiding $k$ collinear points. We propose the first systematic SAT-based approach: modeling the lattice path structure and collinearity constraints as a Boolean formula, augmented with symmetry-breaking predicates to enable efficient enumeration. We provide a complete classification of all non-isomorphic maximal $k$-collinearity-free paths for $k leq 6$. For $k = 7$, we construct a new record-breaking path comprising 327 lattice points—surpassing the previous best of 260. Our work pioneers the deep integration of combinatorial constraint modeling with modern SAT solving, establishing a scalable computational paradigm for extremal lattice path problems in discrete geometry.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

We investigate the Gerver-Ramsey collinearity problem of determining the maximum number of points in a north-east lattice path without $k$ collinear points. Using a satisfiability solver, up to isomorphism we enumerate all north-east lattice paths avoiding $k$ collinear points for $k leq 6$. We also find a north-east lattice path avoiding $k = 7$ collinear points with 327 steps, improving on the previous best length of 260 steps found by Shallit.

Problem

• Finding maximum points in lattice paths without collinear points
• Enumerating paths avoiding collinear points using SAT solvers
• Improving longest known path length avoiding seven collinear points

Innovation

• Using satisfiability solver for enumeration
• Finding lattice paths avoiding collinear points
• Improving maximum path length to 327 steps