## Abstract

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 NP-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 for any given ε ≤ 1. Another topic handled in this paper is the NP-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 3. n is NP-Complete.

Original language | English |
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Pages (from-to) | 33-45 |

Number of pages | 13 |

Journal | Theoretical Computer Science |

Volume | 562 |

Issue number | C |

DOIs | |

State | Published - 2015 |

### Bibliographical note

Publisher Copyright:© 2014 Elsevier B.V.

## Keywords

- Finite alphabets
- NP-Hardness
- Two dimensional tiling

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)