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Binary Pattern Tile Set Synthesis Is NP-hard

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Automata, Languages, and Programming (ICALP 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9134))

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Abstract

We solve an open problem, stated in 2008, about the feasibility of designing efficient algorithmic self-assembling systems which produce 2-dimensional colored patterns. More precisely, we show that the problem of finding the smallest tile assembly system which will self-assemble an input pattern with 2 colors (i.e., \(2\)-Pats) is NP-hard. One crucial lemma makes use of a computer-assisted proof, which is a relatively novel but increasingly utilized paradigm for deriving proofs for complex mathematical problems. This tool is especially powerful for attacking combinatorial problems, as exemplified by the proof for the four color theorem and the recent important advance on the Erdős discrepancy problem using computer programs. In this paper, these techniques will be brought to a new order of magnitude, computational tasks corresponding to one CPU-year. We massively parallelize our program, and provide a full proof of its correctness. Its source code is freely available online.

We thank Manuel Bertrand for his infinite patience and helpful assistance with setting up the server and helping debug our network and system problems, and Cécile Barbier, Eric Fede and Kai Poutrain for their assistance with software setup.

S. Kopecki—Supported by the NSERC Discovery Grant R2824A01 and UWO Faculty of Science grant to L.K.

P.É. Meunier—Supported in part by NSF Grant CCF-1219274.

M.J. Patitz—Supported in part by NSF Grants CCF-1117672 and CCF-1422152.

S. Seki—Supported in part by Academy of Finland, Grant 13266670/T30606.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Western Ontario, London, ON, N6A 1Z8, Canada

    Lila Kari & Steffen Kopecki

  2. Department of Computer Science, Aalto University, P.O.Box 15400, 00076, Aalto, Finland

    Pierre-Étienne Meunier & Shinnosuke Seki

  3. Department of Computer Science and Computer Engineering, University of Arkansas, Fayetteville, AR, USA

    Matthew J. Patitz

  4. Helsinki Institute for Information Technology (HIIT), Espoo, Finland

    Shinnosuke Seki

Authors
  1. Lila Kari
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  2. Steffen Kopecki
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  3. Pierre-Étienne Meunier
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  4. Matthew J. Patitz
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  5. Shinnosuke Seki
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Corresponding author

Correspondence to Matthew J. Patitz .

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Editors and Affiliations

  1. Reykjavik University, Reykjavik, Iceland

    Magnús M. Halldórsson

  2. Kyoto University, Kyoto, Japan

    Kazuo Iwama

  3. The University of Tokyo, Tokyo, Japan

    Naoki Kobayashi

  4. Technische Universiteit Eindhoven, Eindhoven, The Netherlands

    Bettina Speckmann

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Kari, L., Kopecki, S., Meunier, PÉ., Patitz, M.J., Seki, S. (2015). Binary Pattern Tile Set Synthesis Is NP-hard. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_83

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  • DOI: https://doi.org/10.1007/978-3-662-47672-7_83

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