# Complexity of Tiling a Polygon with Trominoes or Bars

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## 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).

## Keywords

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

## Mathematics Subject Classification

52C15 68Q17## Notes

### Acknowledgements

We are grateful to Yoshio Okamoto for his fruitful suggestions, and thank anonymous referees of the preliminary version and of this journal version for their helpful suggestions. This work is partially supported by MEXT/JSPS KAKENHI Grant Numbers JP15K00008 and JP24106007 (T. Horiyama), JP15H00849 and JP16K00004 (T. Ito), JP26730001 (A. Suzuki), and JP26330009 and JP24106004 (R. Uehara).

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