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https://hdl.handle.net/10119/15100
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| Title: | Complexity of Tiling a Polygon with Trominoes or Bars |
| Authors: | Horiyama, Takashi
Ito, Takehiro
Nakatsuka, Keita
Suzuki, Akira
Uehara, Ryuhei |
| Keywords: | tiling problem
polyominoes
NP-complete
#P-complete
ASP-complete |
| Issue Date: | 2017-03-17 |
| Publisher: | Springer |
| Magazine name: | Discrete and Computational Geometry |
| Volume: | 58 |
| Number: | 3 |
| Start page: | 686 |
| End page: | 704 |
| DOI: | 10.1007/s00454-017-9884-9 |
| Abstract: | We study the computational hardness of the tiling puzzle with polyominoes, where a polyomino is a rectilinear polygon (i.e., a polygon made by connecting unit squares.) In the tiling problem, we are given a rectilinear 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 (2001) and Pak and Yang (2013). |
| Rights: | This is the author-created version of Springer, Takashi Horiyama, Takehiro Ito, Keita Nakatsuka, Akira Suzuki, and Ryuhei Uehara, Discrete and Computational Geometry, 58(3), 2017, 686-704. The original publication is available at www.springerlink.com, http://dx.doi.org/10.1007/s00454-017-9884-9 |
| URI: | https://hdl.handle.net/10119/15100 |
| Material Type: | author |
| Appears in Collections: | b10-1. 雑誌掲載論文 (Journal Articles)
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