Tiling simply connected regions with rectangles

Abstract

In [BNRR], it was shown that tiling of general regions with two rectangles is NPcomplete, except for a few trivial special cases. In a different direction, Rémila [Rém2] showed that for simply connected regions by 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 field. 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. We refer to Section 7 for the history of the problem, references, and further remarks.

Rectangular tileability and complementary tileability are undecidable

“…Lemma A.1 lists several classes of Tileability problems that are equivalent. Tiling with rectangles is equivalent with the others for finite regions [PY1], but Theorem 1.4 puts an end to the hope of extending the equivalence to cofinite regions. Here we present an equivalence for special infinite regions, and use it to exhibit yet another undecidable result.…”

“…Here we present an equivalence for special infinite regions, and use it to exhibit yet another undecidable result. We combine an explicit construction in [PY1] with ideas already appearing in this paper. Thus to save space and avoid redundancy, we only sketch the proofs; details will appear in [Yang].…”

“…Proof. Take a set of polyominoes T. The reduction in [PY1] provides a set of rectangles T , and a linear-time function f that transforms a finite region Γ to another finite region Γ , such that Γ is T-tileable if and only if Γ is T -tileable. Explicitly, f takes each unit-length edge of Γ and replaces it with a rectilinear curve to get Γ .…”

“…Tileability on the plane has been a subject of much study [GS]. A lot of focus was in tilings on the square grid by polyominoes [Gol1], including establishing NP-completeness [Lew,MR,PY1] and finding efficient algorithms when possible [KK, Rém]. Aperiodicity in tilings of the entire plane has also been well-studied [Moz, Pen], with connections to ergodic theory [Rad, CR] and quasicrystals [DS].…”

“…Using these methods, Rémila showed that tileability of s.c. regions is quadratic for any two fixed rectangles [Rém]. Perhaps surprisingly, there is a finite set of rectangular tiles such that tileability of s.c. regions is NPcomplete [PY1]. On the other hand, tileability of a rectangular region for a fixed set of tiles can be determined in linear time (see below).…”