A PTAS for the square tiling problem

Amihood Amir; Alberto Apostolico; Gad M. Landau; Oren Sar Shalom

doi:10.1007/978-3-642-16321-0_11

A PTAS for the square tiling problem

Amihood Amir, Alberto Apostolico, Gad M. Landau, Oren Sar Shalom

Department of Computer Science

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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 (1 + ε/2 log n) for any given ε ≤ 1.

Original language	English
Title of host publication	String Processing and Information Retrieval - 17th International Symposium, SPIRE 2010, Proceedings
Pages	118-126
Number of pages	9
DOIs	https://doi.org/10.1007/978-3-642-16321-0_11
State	Published - 2010
Event	17th International Symposium on String Processing and Information Retrieval, SPIRE 2010 - Los Cabos, Mexico Duration: 11 Oct 2010 → 13 Oct 2010

Publication series

Name	Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume	6393 LNCS
ISSN (Print)	0302-9743
ISSN (Electronic)	1611-3349

Conference

Conference	17th International Symposium on String Processing and Information Retrieval, SPIRE 2010
Country/Territory	Mexico
City	Los Cabos
Period	11/10/10 → 13/10/10

Bibliographical note

Funding Information:
Partly supported by NSF grant CCR-09-04581, ISF grant 347/09, and BSF grant 2008217. Partly supported by BSF grant 2008217. Partly supported by the National Science Foundation Award 0904246, Israel Science Foundation grants 35/05 and 347/09, the Israel-Korea Scientific Research Cooperation, Yahoo, Grant No. 2008217 from the United States-Israel Binational Science Foundation (BSF) and DFG.

Funding Information:
★ Partly supported by NSF grant CCR-09-04581, ISF grant 347/09, and BSF grant 2008217. ★★ Partly supported by BSF grant 2008217. ★★★ Partly supported by the National Science Foundation Award 0904246, Israel Sci-ence Foundation grants 35/05 and 347/09, the Israel-Korea Scientific Research Cooperation, Yahoo, Grant No. 2008217 from the United States-Israel Binational Science Foundation (BSF) and DFG.

Funding

Partly supported by NSF grant CCR-09-04581, ISF grant 347/09, and BSF grant 2008217. Partly supported by BSF grant 2008217. Partly supported by the National Science Foundation Award 0904246, Israel Science Foundation grants 35/05 and 347/09, the Israel-Korea Scientific Research Cooperation, Yahoo, Grant No. 2008217 from the United States-Israel Binational Science Foundation (BSF) and DFG. ★ Partly supported by NSF grant CCR-09-04581, ISF grant 347/09, and BSF grant 2008217. ★★ Partly supported by BSF grant 2008217. ★★★ Partly supported by the National Science Foundation Award 0904246, Israel Sci-ence Foundation grants 35/05 and 347/09, the Israel-Korea Scientific Research Cooperation, Yahoo, Grant No. 2008217 from the United States-Israel Binational Science Foundation (BSF) and DFG.

Funders	Funder number
Israel Sci-ence Foundation	35/05
National Science Foundation	CCR-09-04581, 0904246
Bloom's Syndrome Foundation	2008217
Iowa Science Foundation	347/09
Deutsche Forschungsgemeinschaft
United States-Israel Binational Science Foundation
Israel Science Foundation

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10.1007/978-3-642-16321-0_11

Cite this

Amir, A., Apostolico, A., Landau, G. M., & Shalom, O. S. (2010). A PTAS for the square tiling problem. In String Processing and Information Retrieval - 17th International Symposium, SPIRE 2010, Proceedings (pp. 118-126). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6393 LNCS). https://doi.org/10.1007/978-3-642-16321-0_11

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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 (1 + ε/2 log n) for any given ε ≤ 1.",

author = "Amihood Amir and Alberto Apostolico and Landau, {Gad M.} and Shalom, {Oren Sar}",

note = "Funding Information: Partly supported by NSF grant CCR-09-04581, ISF grant 347/09, and BSF grant 2008217. Partly supported by BSF grant 2008217. Partly supported by the National Science Foundation Award 0904246, Israel Science Foundation grants 35/05 and 347/09, the Israel-Korea Scientific Research Cooperation, Yahoo, Grant No. 2008217 from the United States-Israel Binational Science Foundation (BSF) and DFG. Funding Information: ★ Partly supported by NSF grant CCR-09-04581, ISF grant 347/09, and BSF grant 2008217. ★★ Partly supported by BSF grant 2008217. ★★★ Partly supported by the National Science Foundation Award 0904246, Israel Sci-ence Foundation grants 35/05 and 347/09, the Israel-Korea Scientific Research Cooperation, Yahoo, Grant No. 2008217 from the United States-Israel Binational Science Foundation (BSF) and DFG.; 17th International Symposium on String Processing and Information Retrieval, SPIRE 2010 ; Conference date: 11-10-2010 Through 13-10-2010",

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Amir, A, Apostolico, A, Landau, GM & Shalom, OS 2010, A PTAS for the square tiling problem. in String Processing and Information Retrieval - 17th International Symposium, SPIRE 2010, Proceedings. Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 6393 LNCS, pp. 118-126, 17th International Symposium on String Processing and Information Retrieval, SPIRE 2010, Los Cabos, Mexico, 11/10/10. https://doi.org/10.1007/978-3-642-16321-0_11

A PTAS for the square tiling problem. / Amir, Amihood; Apostolico, Alberto; Landau, Gad M. et al.
String Processing and Information Retrieval - 17th International Symposium, SPIRE 2010, Proceedings. 2010. p. 118-126 (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 6393 LNCS).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › peer-review

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Amir A, Apostolico A, Landau GM, Shalom OS. A PTAS for the square tiling problem. In String Processing and Information Retrieval - 17th International Symposium, SPIRE 2010, Proceedings. 2010. p. 118-126. (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)). doi: 10.1007/978-3-642-16321-0_11

A PTAS for the square tiling problem

Abstract

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Bibliographical note

Funding

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