Complexity of Tiling a Polygon with Trominoes or Bars

Takashi Horiyama, Takehiro Ito, Keita Nakatsuka, Akira Suzuki, Ryuhei Uehara

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

We study the computational hardness of the tiling puzzle with polyominoes, where a polyomino is a right-angled polygon (i.e., a polygon made by connecting unit squares along their edges). In the tiling problem, we are given a right-angled polygon P and a set S of polyominoes, and asked whether P can be covered without any overlap using translated copies of polyominoes in S. In this paper, we focus on trominoes and bars as polyominoes; a tromino is a polyomino consisting of three unit squares, and a bar is a rectangle of either height one or width one. Notice that there are essentially two shapes of trominoes, that is, I-shape (i.e., a bar) and L-shape. We consider the tiling problem when restricted to only L-shape trominoes, only I-shape trominoes, both L-shape and I-shape trominoes, or only two bars. In this paper, we prove that the tiling problem remains NP-complete even for such restricted sets of polyominoes. All reductions are carefully designed so that we can also prove the # P-completeness and ASP-completeness of the counting and the another-solution-problem variants, respectively. Our results answer two open questions proposed by Moore and Robson (Discrete Comput Geom 26:573–590, 2001) and Pak and Yang (J Comb Theory 120:1804–1816, 2013).

Original languageEnglish
Pages (from-to)686-704
Number of pages19
JournalDiscrete and Computational Geometry
Volume58
Issue number3
DOIs
Publication statusPublished - 2017 Oct 1

Keywords

  • # P-complete
  • ASP-complete
  • NP-complete
  • Polyominoes
  • Tiling problem

ASJC Scopus subject areas

  • Theoretical Computer Science
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

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