NP-completeness of Tiling Finite Simply Connected Regions with a Fixed Set of Wang Tiles

TL;DR

This paper proves that tiling finite simply connected regions with a fixed set of Wang tiles or rectangles is NP-complete, reducing the number of tiles needed for such complexity results.

Contribution

It establishes NP-completeness for tiling with only 23 Wang tiles and 111 rectangles, improving previous bounds and demonstrating the problem's computational difficulty.

Findings

Tiling with 23 Wang tiles is NP-complete.

Tiling with 111 rectangles is NP-complete.

Number of tiles needed for NP-completeness has been reduced.

Abstract

The computational complexity of tiling finite simply connected regions with a fixed set of tiles is studied in this paper. We show that the problem of tiling simply connected regions with a fixed set of $23$ Wang tiles is NP-complete. As a consequence, the problem of tiling simply connected regions with a fixed set of $111$ rectangles is NP-complete. Our results improve that of Igor Pak and Jed Yang by using fewer numbers of tiles. Notably in the case of Wang tiles, the number has decreased by more than one third from $35$ to $23$.