Hard and Easy Instances of L-Tromino Tilings

TL;DR

This paper investigates the computational complexity of tiling regions with L-shaped trominoes, identifying cases where the problem is NP-complete or solvable in polynomial time, depending on region restrictions and tromino orientations.

Contribution

It characterizes tiling possibilities for Aztec shapes and introduces polynomial algorithms for regions without specific forbidden sub-regions under certain conditions.

Findings

Tiling Aztec rectangles and diamonds with L-trominoes can be characterized.

Deciding tilings with only 180° rotations of L-trominoes is NP-complete.

Polynomial algorithms exist for regions excluding certain forbidden polyominoes.

Abstract

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 and an Aztec diamond has an L-tromino tiling. Then, we study tilings of arbitrary regions where only $180^\circ$ 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 does not contains certain so-called "forbidden polyominoes" as sub-regions, then there exists a polynomial time algorithm for deciding a tiling.