ASP-Completeness of Hamiltonicity in Grid Graphs, with Applications to Loop Puzzles

TL;DR

The paper establishes ASP-completeness for Hamiltonicity in grid graphs, proving that Hamiltonicity in max-degree-$3$ grid graphs (directed or undirected) is ASP-complete and extending the same result to directed rectangular grids and certain edge-subset variants. It develops a unifying reduction framework based on Tree-Residue Vertex-Breaking (TRVB) and a versatile T-metacell gadget system, enabling parsimonious encodings of degree-$3$ vertices within grid structures. The authors apply this framework to a broad suite of pencil-and-paper loop puzzles (38 shown), deriving ASP-completeness and, in many cases, novel NP-hardness results, thereby substantially expanding the catalog of ASP-hard puzzle variants. Together, these results furnish a flexible, parsimony-preserving toolkit for proving ASP-completeness in grid-based problems and offer practical guidance for the complexity classification of loop-drawing puzzles.

Abstract

We prove that Hamiltonicity in maximum-degree-3 grid graphs (directed or undirected) is ASP-complete, i.e., it has a parsimonious reduction from every NP search problem (including a polynomial-time bijection between solutions). As a consequence, given k Hamiltonian cycles, it is NP-complete to find another; and counting Hamiltonian cycles is #P-complete. If we require the grid graph's vertices to form a full $m \times n$ rectangle, then we show that Hamiltonicity remains ASP-complete if the edges are directed or if we allow removing some edges (whereas including all undirected edges is known to be easy). These results enable us to develop a stronger "T-metacell" framework for proving ASP-completeness of rectangular puzzles, which requires building just a single gadget representing a degree-3 grid-graph vertex. We apply this general theory to prove ASP-completeness of 38 pencil-and-paper puzzles where the goal is to draw a loop subject to given constraints: Slalom, Onsen-meguri, Mejilink, Detour, Tapa-Like Loop, Kouchoku, Icelom; Masyu, Yajilin, Nagareru, Castle Wall, Moon or Sun, Country Road, Geradeweg, Maxi Loop, Mid-loop, Balance Loop, Simple Loop, Haisu, Reflect Link, Linesweeper; Vertex/Touch Slitherlink, Dotchi-Loop, Ovotovata, Building Walk, Rail Pool, Disorderly Loop, Ant Mill, Koburin, Mukkonn Enn, Rassi Silai, (Crossing) Ichimaga, Tapa, Canal View, Aqre, and Paintarea. The last 14 of these puzzles were not even known to be NP-hard. Along the way, we prove ASP-completeness of some simple forms of Tree-Residue Vertex-Breaking (TRVB), including planar multigraphs with degree-6 breakable vertices, or with degree-4 breakable and degree-1 unbreakable vertices.

Figures (14)

• Figure 1: Illustration of Lemma \ref{['lem:connections']}.
• Figure 2: The bijections we define for Lemma \ref{['lem:connections']}.
• Figure 3: Simulating a degree-4 unbreakable vertex using degree-4 breakable vertices (white) and degree-1 unbreakable vertices (black).
• Figure 4: Simulating a degree-4 unbreakable vertex using degree-6 breakable vertices.
• Figure 5: An example showing how reductions from TRVB to Hamiltonian cycle work.

Theorems & Definitions (40)

• Definition 2.1: trvb
• Definition 2.2: BENT1987barnette
• Theorem 2.3
• Lemma 2.4
• proof
• Lemma 2.5
• proof
• Lemma 2.7
• proof
• Theorem 3.1