Binary Pattern Tile Set Synthesis Is NP-hard

  • Lila Kari
  • Steffen Kopecki
  • Pierre-Étienne Meunier
  • Matthew J. Patitz
  • Shinnosuke Seki
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)


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.


Input Pattern Conjunctive Normal Form Tile Type Rectangular Pattern Discrepancy Conjecture 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Lila Kari
    • 1
  • Steffen Kopecki
    • 1
  • Pierre-Étienne Meunier
    • 2
  • Matthew J. Patitz
    • 3
  • Shinnosuke Seki
    • 2
    • 4
  1. 1.Department of Computer ScienceUniversity of Western OntarioLondonCanada
  2. 2.Department of Computer ScienceAalto UniversityAaltoFinland
  3. 3.Department of Computer Science and Computer EngineeringUniversity of ArkansasFayettevilleUSA
  4. 4.Helsinki Institute for Information Technology (HIIT)EspooFinland

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