Hard and Easy Instances of L-Tromino Tilings

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 indentify restrictions to the problem where either it remains NP-complete or it has a polynomial time algorithm. First we show that an aztec diamond of order n always has an L-tromino tiling if and only if n(n+1) ≡ 0 mod 3; if an aztec diamond has at least two 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 so-called “forbidden polyominoes” as subregions, then there exists a polynomial time algorithm for deciding a tiling.