Journal of Combinatorial Optimization

, Volume 33, Issue 2, pp 496–529 | Cite as

A manually-checkable proof for the NP-hardness of 11-color pattern self-assembly tileset synthesis

  • Aleck Johnsen
  • Ming-Yang Kao
  • Shinnosuke Seki


Patterned self-assembly tile set synthesis (pats) aims at minimizing the number of distinct DNA tile types used to self-assemble a given rectangular color pattern. For an integer kk-pats is the subproblem of pats that restricts input patterns to those with at most k colors. We give an efficient Open image in new window verifier, and based on that, we establish a manually-checkable proof for the NP-hardness of 11-pats; the best previous manually-checkable proof is for 29-pats.


DNA pattern self-assembly Tile complexity Manually-checkable proof 



We are very thankful to the anonymous referees for their valuable comments on the earlier versions of this paper. This work is supported in part by NSF Grants CCF-1049899 and CCF-1217770 to M-Y. Kao and by HIIT Pump Priming Grant 902184/T30606, Academy of Finland, Postdoctoral Researcher Grant 13266670/T30606, JST Program to Disseminate Tenure Tracking System, MEXT, JAPAN, No. 6F36, and JSPS Grant-in-Aid for Research Activity Start-up No. 15H06212 to S. Seki.


  1. Barish R, Rothemund PWK, Winfree E (2005) Two computational primitives for algorithmic self-assembly: copying and counting. Nano Lett 5(12):2586–2592CrossRefGoogle Scholar
  2. Brun Y (2008) Solving NP-complete problems in the tile assembly model. Theor Comput Sci 395:31–46MathSciNetCrossRefzbMATHGoogle Scholar
  3. Brun Y (2008) Solving satisfiability in the tile assembly model with a constant-size tileset. J Algorithms 63(4):151–166MathSciNetCrossRefzbMATHGoogle Scholar
  4. Brun Y (2012) Efficient 3-SAT algorithms in the tile assembly model. Nat Comput 11:209–229MathSciNetCrossRefzbMATHGoogle Scholar
  5. Cook M, Rothemund PWK, Winfree E (2004) Self-assembled circuit patterns. In: Proceedings of the 9th International Workshop on DNA Based Computers (DNA 9), LNCS, vol. 2943. Springer, p 91–107Google Scholar
  6. Culik K, Kari J (1997) On aperiodic sets of Wang tiles. Foundations of computer science potential—theory—cognition, LNCS, vol 1337. Springer, Berlin, pp 153–162CrossRefGoogle Scholar
  7. Czeizler E, Popa A (2013) Synthesizing minimal tile sets for complex patterns in the framework of patterned DNA self-assembly. Theor Comput Sci 499:23–37MathSciNetCrossRefzbMATHGoogle Scholar
  8. Evans CG (2014) Crystals that count! physical principles and experimental investigations of DNA tile self-assembly. Ph.D. thesis, California Institute of TechnologyGoogle Scholar
  9. Göös M, Lempiäinen T, Czeizler E, Orponen P (2014) Search methods for tile sets in patterned DNA self-assembly. J Comput Syst Sci 80:297–319MathSciNetCrossRefzbMATHGoogle Scholar
  10. Johnsen A, Kao MY, Seki S (2013) Computing minimum tile sets to self-assemble patterns in 29-colors. In: Proceedings of the 24th International Symposium on Algorithms and Computation (ISAAC 2013), LNCS, vol. 8283. Springer, p 699–710Google Scholar
  11. Kari L, Kopecki S, Étienne Meunier P, Patitz MJ, Seki S (2015a) Binary pattern tile set synthesis is NP-hard. In: Proceedings of the 42nd International Colloquium on Automata, Languages, and Programming (ICALP 2015), LNCS, vol. 9134. Springer, p 1022–1034Google Scholar
  12. Kari L, Kopecki S, Seki S (2015b) 3-color bounded patterned self-assembly. Nat Comput 14(2):279–292Google Scholar
  13. Ma X, Lombardi F (2008) Synthesis of tile sets for DNA self-assembly. IEEE Trans Comput-Aided Des Integr Circuits Syst 27(5):963–967CrossRefGoogle Scholar
  14. Ma X, Lombardi F (2009) On the computational complexity of tile set synthesis for DNA self-assembly. IEEE Trans Circuits Syst II 56(1):31–35CrossRefGoogle Scholar
  15. Rothemund PWK, Papadakis N, Winfree E (2004) Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol 2(12):e424CrossRefGoogle Scholar
  16. Schaefer TJ (1978) The complexity of satisfiability problems. In: Proceedings of the 10th Annual ACM Symposium on Theory of Computing (STOC 1978), p 216–226Google Scholar
  17. Seki S (2013) Combinatorial optimization in pattern assembly (extended abstract). In: Proceedings of the 12th International Conference on Unconventional Computation and Natural Computation (UCNC 2013), LNCS, vol. 7956. Springer, p 220–231Google Scholar
  18. Stefanovic D, Turberfield A (eds) (2012) The 18th International Conference on DNA Computing and Molecular Programming, Aarhus, Denmark, 14–17 August 2012Google Scholar
  19. Winfree E (1998) Algorithmic self-assembly of DNA. Ph.D. thesis, California Institute of TechnologyGoogle Scholar
  20. Winfree E (2000) Algorithmic self-assembly of DNA: theoretical motivations and 2D assembly experiments. J Biomol Struct Dyn Special Issue S2:263–270CrossRefGoogle Scholar
  21. Winfree E, Liu F, Wenzler LA, Seeman NC (1998) Design and self-assembly of two-dimensional DNA crystals. Nature 394:539–544CrossRefGoogle Scholar
  22. Zhang J, Liu Y, Ke Y, Yan H (2006) Periodic square-like gold nanoparticle arrays templated by self-assembled 2D DNA nanogrids on a surface. Nano Lett 6(2):248–251CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Electrical Engineering and Computer ScienceNorthwestern UniversityEvanstonUSA
  2. 2.Helsinki Institute for Information Technology (HIIT) and Department of Computer ScienceAalto UniversityAaltoFinland
  3. 3.Department of Communication Engineering and InformaticsThe University of Electro-CommunicationsChofuJapan

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