Required-edge Cycle Cover Problem: an ASP-Completeness Framework for Graph Problems and Puzzles

TL;DR

A flow model equivalent to the constrained RCCP is introduced; this model allows gadgets to be tiled densely on a rectangular grid, which enables RCCP to various pencil-and-paper puzzles in a parsimonious manner, and proves the ASP-completeness of several puzzles.

Abstract

Proving the NP-completeness of pencil-and-paper puzzles typically relies on reductions from combinatorial problems such as the satisfiability problem (SAT). Although the properties of these problems are well studied, their purely combinatorial nature often does not align well with the geometric constraints of puzzles. In this paper, we introduce the Required-edge Cycle Cover Problem (RCCP) -- a variant of the Cycle Cover Problem (CCP) on mixed graphs. CCP on mixed graphs was studied by Seta (2002) to establish the ASP-completeness (i.e., NP-completeness under parsimonious reductions) of the puzzle Kakuro (a.k.a.~Cross Sum), and is known to be ASP-complete under certain conditions. We prove the ASP-completeness of RCCP under certain conditions, and strengthen known ASP-completeness results of CCP on mixed graphs as a corollary. Using these results, we resolve the ASP-completeness of Constraint Graph Satisfiability (CGS) in a certain case, addressing an open problem posed by the MIT Hardness Group (2024). We also show that Kakuro remains ASP-complete even when the available digit set is $\{1, 2, 3\}$, consequently completing its complexity classification regarding the maximum available digit and the maximum lengths of contiguous blank cells. It strengthens previously known results of Seta (2002) and Ruepp and Holzer (2010). Furthermore, we introduce a flow model equivalent to the constrained RCCP; this model allows gadgets to be tiled densely on a rectangular grid, which enables us to reduce RCCP to various pencil-and-paper puzzles in a parsimonious manner. By applying this framework, we prove the ASP-completeness of several puzzles, including Chocona, Four Cells, Hinge, and Shimaguni, and strengthen existing NP-completeness results for Choco Banana and Five Cells to ASP-completeness.

Figures (23)

• Figure 1: An example of a variable gadget and its two valid cycle covers. Each pair of open edges indicates connection to a clause gadget. While this gadget consists of three connections, it can be extended to any number, corresponding to the degree of the variable in the formula.
• Figure 2: Top row: the clause gadget. Bottom row: its three valid cycle covers. Three pairs of open edges indicate connections to variable gadgets. The terminal of each connection consists of two vertices and a required edge between them. In each valid configuration, exactly two terminals of connections are covered, enforcing that exactly one variable is assigned true.
• Figure 3: Simulation of a required edge using undirected and directed edges.
• Figure 4: An example of the flow model construction. Labels on the sources of $N$ indicate their types: $u$ for unconstrained, $b$ for biased (toward the arrow), $x$ for exclusive, and $f$ for fixed.
• Figure 6: Three types of vertices in constraint graphs and all of their valid orientations.

Theorems & Definitions (22)

• Definition 1
• Definition 2
• Definition 3
• Theorem 4: Seta02CrossSum
• Definition 5
• Theorem 6
• Corollary 7
• Remark 8
• Definition 9
• Lemma 9