Natural Computing

, Volume 14, Issue 2, pp 279–292 | Cite as

3-color bounded patterned self-assembly

  • Lila Kari
  • Steffen Kopecki
  • Shinnosuke Seki


The problem of patterned self-assembly tile set synthesis (Pats) is to find a minimal tile set which uniquely self-assembles into a given pattern. Czeizler and Popa proved the \(\mathrm {NP}\)-completeness of Pats and Seki showed that the Pats problem is already \(\mathrm {NP}\)-complete for patterns with 60 colors. In search for the minimal number of colors such that Pats remains \(\mathrm {NP}\)-complete, we introduce multiple bound Pats (mbPats) where we allow bounds for the numbers of tile types of each color. We show that mbPats is \(\mathrm {NP}\)-complete for patterns with just three colors and, as a byproduct of this result, we also obtain a novel proof for the \(\mathrm {NP}\)-completeness of Pats which is more concise than the previous proofs.


Conjunctive Normal Form Boolean Formula Tile Type Unique Color Shaped Seed 
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Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  1. 1.Department of Computer ScienceThe University of Western OntarioLondonCanada
  2. 2.Department of Information and Computer Science, Helsinki Institute for Information Technology (HIIT)Aalto UniversityAaltoFinland

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