On Subtilings of Polyomino Tilings

TL;DR

This paper investigates the complexity of subdividing polyomino tilings of rectangles into smaller rectangles and establishes NP-completeness for certain cases while providing bounds for when subtilings always exist.

Contribution

It proves NP-completeness of the subtiling decision problem for rectangular polyominoes and identifies conditions under which subtilings are guaranteed to exist.

Findings

NP-complete decision problem for rectangular polyominoes

Existence of subtilings for large enough rectangles

Bounds on rectangle sizes for guaranteed subtilings

Abstract

We consider a problem concerning tilings of rectangular regions by a finite library of polyominoes. We specifically look at rectangular regions of dimension $n\times m$ and ask whether or not a tiling of this region can be rearranged so that tiling of the $n\times m$ rectangle can be realized as a tiling of an $n\times m'$ rectangle and an $n\times m"$ rectangle, $m=m'+m"$. We call this a subtiling. We show that the associated decision problem is $\mathsf{NP}$-complete when restricted to rectangular polyominoes. We also show that for certain finite libraries of polyominoes, if $m$ is sufficiently large, a subtiling always exists and give bounds.