Hard and easy instances of L-tromino tilings

Abstract

In this work we study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we identify restrictions to the problem where it either remains NP-complete or has a polynomial time algorithm. First, we characterize the possibility of when an Aztec rectangle has an L-tromino tiling, and hence also an Aztec diamond; if an Aztec rectangle has an unknown number of defects or holes, however, the problem of deciding a tiling is NP-complete. Then, we study tilings of arbitrary regions where only 180 • rotations of L-trominoes are available. For this particular case we show that deciding the existence of a tiling remains NP-complete; yet, if a region contains certain socalled "forbidden polyominoes" as subregions, then there exists a polynomial time algorithm for deciding a tiling.

On Enumeration and Entropy of Ribbon Tilings

“…First, note that usual Aztec diamonds are non-tileable by n-ribbons if n 3. (For the case n = 3, see Theorem 17 in [AGMSV20].) For this reason, we consider generalized Aztec diamonds which are tileable by n-ribbons.…”

“…The algorithm is linear in the area of the region. In [AGMSV20], it was shown that for general regions (which are allowed to be non-simply connected with arbitrary number of holes), the existence of tilings by 180-trominoes is an N P -complete decision problem.…”

“…Dominoes are a particular case of ribbon tiles with n = 2. Ribbon tiles with n = 3 are called (right-oriented) 180-trominoes in [AGMSV20]. 2 The study of ribbon tilings for n 3 was initiated in [Pak00], and developed extensively in [Sheffield02].…”

Tromino Tilings with Pegs via Flow Networks

“…There is a natural correspondence between domino tilings and perfect matchings in dual graphs. Likewise, Akagi et al [1] showed that tromino tilings correspond to independent sets in a slight variation of dual graphs called an intersection graph.…”

“…Horiyama et al [6] strengthened the NP-completeness result of Moore and Robson and proved that counting the number of tromino packings of a region is #P-complete. More recently, Akagi et al [1] showed that the NP-completeness of deciding a tromino tiling heavily depends on geometrical properties of the region; for example, a region with the shape of an Aztec rectangle admits a polynomial time algorithm that decides a tromino tiling, whereas the problem remains NP-complete for an Aztec rectangle with at least two holes.…”