Tiling simply connected regions with rectangles

Igor Pak, Jed Yang
2013 Journal of combinatorial theory. Series A  
In [BNRR], it was shown that tiling of general regions with two rectangles is NP-complete, except for few trivial special cases. In a different direction, Rémila [Rém2] showed that for simply connected regions and two rectangles, the tileability can be solved in quadratic time (in the area). We prove that there is a finite set of at most 10 6 rectangles for which the tileability problem of simply connected regions is NP-complete, closing the gap between positive and negative results in the
more » ... . We also prove that counting such rectangular tilings is #P-complete, a first result of this kind. Theorem 1.2 There exists a finite set R of at most 10 6 rectangular tiles, such that counting the number of tilings of simply connected regions with R is #P-complete. Although #P-completeness is known for tilings of general regions with right tromino and square tetromino [MR], nothing was known for tilings with rectangles, or for tilings of simply connected regions. We refer to Section 8 for the history of the problem, references, and further remarks.
doi:10.1016/j.jcta.2013.06.008 fatcat:3ihbnp43drgojopcevxojzfzgi