NP-Completeness Proofs of Puzzles using the T-Metacell Framework

TL;DR

The paper investigates the $NP$- and $ASP$-hardness of pencil puzzles through the T-metacell framework, which encodes grid-graph Hamiltonicity with degree-3 gadgets. It shows three puzzles—Grand Tour, Entry-Exit, and Zahlenschlange—are $ASP$-complete, and a fourth puzzle, Yagit, is $NP$-complete, using reductions that rely on Hamiltonian-cycle encodings and carefully designed forced-edge and asymmetric gadgets. The results underscore the versatility of the T-metacell approach in proving strong hardness classifications for constraint-based puzzles. These contributions provide a unified methodology for establishing complexity classifications across diverse pencil-puzzle systems and pave the way for further applications of the framework.

Abstract

Pencil puzzles are puzzles that can be solved by writing down solutions on a paper, using only logical reasoning. In this paper, we utilize the "T-metacell" framework developed by Tang and the MIT Hardness Group to prove the NP-completeness of four new pencil puzzles: Grand Tour, Entry Exit, Zahlenschlange, and Yagit. Additionally, the first three are also proven to be ASP-complete. The results demonstrate how versatile the framework is, offering new insights into the computational complexity of problems with various constraints.

Figures (20)

• Figure 1: 5×5 undirected T-metacell for Grand Tour with all exit choices
• Figure 2: 9x9 and 17x17 undirected T-metacell for Grand Tour with all exit choices
• Figure 3: 5x5 Asymmetrical Forced-edge Undirected T-metacell
• Figure 4: 9x9 forced-edge undirected T-metacell for Entry-Exit step-by-step traversal
• Figure 5: $n^{th}$ Asymmetrical Forced-edge Undirected T-metacell