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

TL;DR

This paper introduces the Required-edge Cycle Cover Problem (RCCP), proves its ASP-completeness, and uses it to establish the complexity of various puzzles, advancing the understanding of their computational hardness.

Contribution

It establishes the ASP-completeness of RCCP, strengthens known results on Kakuro, and provides a unified flow model to prove puzzle complexities.

Findings

RCCP is ASP-complete under certain conditions.

Kakuro remains ASP-complete with limited digits and cell lengths.

Several puzzles are proven ASP-complete using the flow model.

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.