The Troubles of Interior Design–A Complexity Analysis of the Game Heyawake

Abstract

Heyawake is one of many recently popular Japanese pencil puzzles. We investigate the computational complexity of the problem of deciding whether a given puzzle instance has a solution or not. We show that Boolean gates can be emulated via Heyawake puzzles, and that it is possible to reduce the Boolean Satisfiability problem to Heyawake. It follows that the problem in question is NP-complete.

Playing games with algorithms: Algorithmic combinatorial game theory

“…In Hiroimono or Goishi Hiroi [And07], we are given a collection of stones at vertices of a rectangular grid, and the goal is to find a path that visits all stones, changes directions by ±90 • and only at stones, and removes stones as they are visited (similar to Phutball in Section 4.15). In Heyawake [HR07], we are given a subdivision of a rectangular grid into rectangular rooms, some of which are labeled with a positive integer, and the goal is to paint a subset of unit squares so that the number of painted squares in each labeled room equals the label, painted squares are never (horizontally or vertically) adjacent, unpainted squares are connected (via horizontal and vertical connections), and maximal contiguous (horizontal or vertical) strips of squares intersect at most two rooms. In Hitori [Hea08c], we are given a rectangular grid with each square labeled with an integer, and the goal is to paint a subset of unit squares so that every row and every column has no repeated unpainted label (similar to Sudoku), painted squares are never (horizontally or vertically) adjacent, and unpainted squares are connected (via horizontal and vertical connections).…”

Note on Algorithmic Investigations of Juosan Puzzles

“…Studies regarding this include algorithmic investigations and computational complexity analysis of various puzzles (see [7][8][9][10] for extensive surveys). In the case of penciland-paper puzzles, many have been proven to be NPcomplete, such as (in chronological order): Nonogram (1996) [11], Sudoku (2003) [12], Nurikabe (2004) [13], Heyawake (2007) [14], Country Road and Yajilin (2012) [15], Kurodoko (2012) [16], Sto-20 Jurnal Ilmu Komputer dan Informasi (Journal of Computer Science and Information), volume 17, issue 1, February 2024…”

Solving Yin-Yang Puzzles Using Exhaustive Search and Prune-and-Search Algorithms

“…Several systematic studies have been carried out on the topic of the complexity of puzzles [3]- [5]. Moreover, many pencil-andpaper based puzzles have been proven to be NP-complete, such as (in chronological order): Sudoku (2003) [6], Nurikabe (2004) [7], Hiroimono (2007) [8], Heyawake (2007) [9], Hashiwokakero (2009) [10], Kurodoko (2012) [11], Yajilin and Country Road (2012) [12], Shikaku and Ripple Effect (2013) [13], Yosenabe (2014) [14], Shakashaka (2014) [15], Fillmat (2015) [16], Usowan (2018) [17], Sto-Stone (2018) [18], Dosun-Fuwari (2018), [19], Tatamibari (2020) [20], Kurotto and Juosan (2020) [21], and Yin-Yang (2021) [1].…”