Col is PSPACE-complete on Triangular Grids

2025-01-11

AI Summary

This paper investigates the computational complexity of the combinatorial game Col on triangular grid graphs, establishing that its winnability problem is PSPACE-complete. We achieve this via a polynomial-time reduction from Bounded 2-Player Constraint Logic, the first such hardness proof for Col on triangular grids—a highly symmetric, regular geometric graph family. Our construction rigorously encodes constraint logic gadgets within the structural constraints of triangular lattices, preserving game equivalence under optimal play. This result resolves an open question regarding Col’s complexity on geometrically structured graphs, filling a key theoretical gap in the complexity landscape of positional games. Moreover, triangular grids constitute the most regular and geometrically uniform graph family known to be PSPACE-hard for Col, surpassing previously identified instances in structural regularity. The work thus advances the classification of combinatorial games on structured graphs, providing both a new hardness benchmark and methodological insights for complexity analysis of games under geometric constraints.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Game Theory and Economic Paradigms: Cooperative Game Theory

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Security and Privacy: Large-scale security measurements

Abstract

We demonstrate that Col is PSPACE-complete on triangular grid graphs via a reduction from Bounded Two-Player Constraint Logic. This is the most structured graph family that Col is known to be computationally hard for.

Problem

• PSPACE-complete
• Combinatorial game theory
• Triangle mesh

Innovation

• PSPACE-Complete
• Col Game
• Triangular Mesh