A PTAS for the Square Tiling Problem

The Square Tiling Problem was recently introduced as equivalent to the problem of reconstructing an image from patches and a possible general-purpose indexing tool. Unfortunately, the Square Tiling Problem was shown to be NPNP-hard. A 1/2-approximation is known.We show that if the tile alphabet is fixed and finite, there is a Polynomial Time Approximation Scheme (PTAS) for the Square Tiling Problem with approximation ratio of (1−ϵ2log⁡n) for any given ϵ≤1ϵ≤1.Another topic handled in this paper is the NPNP-hardness of the Tiling problem with an infinite alphabet. We show that when the alphabet is not bounded, even the decision version for rectangles of size 3n   is NPNP-Complete.