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

Pakapim Eua-anant, Papangkorn Apinyanon, Thunyatorn Jirachaisri, Nantapong Ruangsuksriwong, Suthee Ruangwises

TL;DR

This paper proves that All or Nothing and Water Walk are NP-complete using the T-metacell framework, reducing from Hamiltonian cycle problems on a max-degree-3 spanning subgraph of a rectangular grid graph, denoting the base graph as $G$ with complement $H$. The reductions encode grid-graph vertices with T-metacell gadgets and rely on representing the base graph via $G$ and $H$, without requiring unique local paths. Because the gadgets do not enforce a unique path between exit pairs, the results establish NP-completeness but not ASP-completeness, leaving ASP as a target for future work. Overall, the work expands the catalog of pencil puzzles known to be NP-hard and outlines a concrete path toward achieving ASP-completeness through stronger locally-unique gadgets.

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.

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

TL;DR

This paper proves that All or Nothing and Water Walk are NP-complete using the T-metacell framework, reducing from Hamiltonian cycle problems on a max-degree-3 spanning subgraph of a rectangular grid graph, denoting the base graph as with complement . The reductions encode grid-graph vertices with T-metacell gadgets and rely on representing the base graph via and , without requiring unique local paths. Because the gadgets do not enforce a unique path between exit pairs, the results establish NP-completeness but not ASP-completeness, leaving ASP as a target for future work. Overall, the work expands the catalog of pencil puzzles known to be NP-hard and outlines a concrete path toward achieving ASP-completeness through stronger locally-unique gadgets.

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.
Paper Structure (8 sections, 3 theorems, 8 figures)

This paper contains 8 sections, 3 theorems, 8 figures.

Table of Contents

  1. Introduction
  2. T-Metacell Framework
  3. NP-Completeness and ASP-Completeness
  4. Our Contribution
  5. NP-Completeness Proofs
  6. All or Nothing
  7. Water Walk
  8. Conclusion and Future Work

Key Result

Lemma 1

The one-cell region inside every T-metacell is dead.

Figures (8)

  • Figure 1: Graphs $G$, $H$, and the corresponding graph $G'$ constructed from $G$ using T-metacells
  • Figure 2: An All or Nothing instance (left) and one of its solutions (right)
  • Figure 3: A T-metacell gadget, with the red lines indicating region boundaries, and the shaded area indicating foundational parts and the one-cell region
  • Figure 4: Path between each pair of exits shown in blue
  • Figure 5: $+$-type cells and $?$-type cells in the foundational parts of a T-metacell
  • ...and 3 more figures

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof