The Complexity of Generalized Domino Tilings

Pak, Igor; Yang, Jed

doi:10.37236/2554

Cited by 12 publications

(9 citation statements)

References 25 publications

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“…There are instances where results of tilings in the plane do not extend to higher dimensions. For example, the number of tilings of a finite region by dominoes can be counted efficiently in the plane, but is #P-complete in higher dimensions [PY2]. However, all of the results in this paper extend easily to higher dimensions.…”

Section: Final Remarks and Open Problemsmentioning

confidence: 93%

Rectangular tileability and complementary tileability are undecidable

Yang

2014

European Journal of Combinatorics

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Does a given a set of polyominoes tile some rectangle? We show that this problem is undecidable. In a different direction, we also consider tiling a cofinite subset of the plane. The tileability is undecidable for many variants of this problem. However, we present an algorithm for testing whether the complement of a finite region is tileable by a set of rectangles.We discuss this connection and some curious consequences of Theorem 1.2 in Subsection 7.2. It is worth noting that if we are given a finite region to tile as opposed to the entire plane, the

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Section: Final Remarks and Open Problemsmentioning

confidence: 93%

Rectangular tileability and complementary tileability are undecidable

Yang

2014

European Journal of Combinatorics

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“…the previous case, t is equivalent to a product of copies of t +1 and t −1 , completing this last case and the proof of the lemma. 10. Regularity of boxes for n = 4…”

Section: Domino Tilings In Higher Dimensionsmentioning

confidence: 99%

Domino tilings and flips in dimensions 4 and higher

Klivans¹,

Saldanha²

2022

Algebraic Combinatorics

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In this paper we consider domino tilings of bounded regions in dimension n 4. We define the twist of such a tiling, an elements of Z/(2), and prove that it is invariant under flips, a simple local move in the space of tilings.We investigate which regions D are regular, i.e. whenever two tilings t 0 and t 1 of D × [0, N ] have the same twist then t 0 and t 1 can be joined by a sequence of flips provided some extra vertical space is allowed. We prove that all boxes are regular except D = [0, 2] 3 . Furthermore, given a regular region D, we show that there exists a value M (depending only on D) such that if t 0 and t 1 are tilings of equal twist of D × [0, N ] then the corresponding tilings can be joined by a finite sequence of flips in D × [0, N + M ]. As a corollary we deduce that, for regular D and large N , the set of tilings of D × [0, N ] has two twin giant components under flips, one for each value of the twist.

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“…Bodini [1] considered tileability problems of pyramidal polycubes. Pak and Yang [14] studied the complexity of the problems of tileability and counting for domino tilings in three and higher dimensions, and proved some hardness results in this respect. If R is a cubiculated region in R 3 , a floor of R is R ∩ (R 2 × [n, n + 1]), for some n ∈ Z.…”

Section: Introductionmentioning

confidence: 99%

Flip Invariance for Domino Tilings of Three-Dimensional Regions with Two Floors

Milet

Saldanha

2015

Discrete Comput Geom

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We investigate tilings of cubiculated regions with two simply connected floors by 2 × 1 × 1 bricks. More precisely, we study the flip connected component for such tilings, and provide an algebraic invariant that "almost" characterizes the flip connected components of such regions, in a sense that we discuss in the paper. We also introduce a new local move, the trit, which, together with the flip, connects the space of domino tilings when the two floors are identical.

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The Complexity of Generalized Domino Tilings

Abstract: Tiling planar regions with dominoes is a classical problem in which the decision and counting problems are polynomial. We prove a variety of hardness results (both NP-and #Pcompleteness) for different generalizations of dominoes in three and higher dimensions.

Cited by 12 publications

References 25 publications

Rectangular tileability and complementary tileability are undecidable

Rectangular tileability and complementary tileability are undecidable

Domino tilings and flips in dimensions 4 and higher

Flip Invariance for Domino Tilings of Three-Dimensional Regions with Two Floors

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