Rectangular tileability and complementary tileability are undecidable

TL;DR

This paper proves that determining whether a set of polyominoes can tile a rectangle or a cofinite subset of the plane is undecidable, but provides an algorithm for a specific rectangle tiling problem.

Contribution

It establishes the undecidability of rectangular and complementary tileability problems and introduces an algorithm for a particular finite-region complement tiling problem.

Findings

Tileability of rectangles by polyominoes is undecidable.

Tileability of cofinite plane subsets is undecidable for many variants.

An algorithm exists for testing tileability of the complement of a finite region by rectangles.

Abstract

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.