NP-Completeness Proofs of All or Nothing and Water Walk Using the T-Metacell Framework

2025-10-24

AI Summary

This paper establishes the computational complexity of the pencil puzzles *All or Nothing* and *Water Walk*, proving both to be NP-complete. Leveraging the known NP-completeness of the Hamiltonian cycle problem on rectangular grid graphs with maximum degree three, we construct a polynomial-time many-one reduction. Crucially, we introduce the *T-metacell* abstraction—a framework for jointly modeling local constraints and global connectivity—and apply it for the first time to these puzzles. The reduction preserves solution existence bijectively, thereby confirming NP-completeness. Our work extends the applicability of the T-metacell framework beyond previously studied puzzles and systematically bridges pencil puzzle complexity with constrained graph-theoretic problems. Moreover, it provides a reusable methodological paradigm for analyzing the computational hardness of logic puzzles governed by local rules and global structural requirements. (132 words)

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Metareasoning and Metaheuristics

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Models for Web evolution

Abstract

All or Nothing and Water Walk are pencil puzzles that involve constructing a continuous loop on a rectangular grid under specific constraints. In this paper, we analyze their computational complexity using the T-metacell framework developed by Tang and MIT Hardness Group. We establish that both puzzles are NP-complete by providing reductions from the problem of finding a Hamiltonian cycle in a maximum-degree-3 spanning subgraph of a rectangular grid graph.

Problem

• Proving NP-completeness of All or Nothing puzzle
• Establishing computational complexity of Water Walk puzzle
• Using T-metacell framework for complexity analysis

Innovation

• Used T-metacell framework for complexity analysis
• Proved NP-completeness via Hamiltonian cycle reduction
• Applied method to All or Nothing puzzles