Algorithmica

, Volume 78, Issue 1, pp 1–46 | Cite as

Binary Pattern Tile Set Synthesis Is NP-Hard

  • Lila Kari
  • Steffen Kopecki
  • Pierre-Étienne Meunier
  • Matthew J. Patitz
  • Shinnosuke Seki
Article
  • 296 Downloads

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 rectilinearly self-assembles an input pattern with 2 colors (i.e., 2-Pats) is \(\mathbf {NP}\)-hard. Of both theoretical and practical significance, the more general k-Pats problem has been studied in a series of papers which have shown k-Pats to be \(\mathbf {NP}\)-hard for \(k=60\), \(k=29\), and then \(k=11\). In this paper, we prove the fundamental conjecture that 2-Pats is \(\mathbf {NP}\)-hard, concluding this line of study. While most of our proof relies on standard mathematical proof techniques, 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.

Keywords

Algorithmic DNA self-assembly Pattern assembly NP-hardness Computer-assisted proof Massively-parallelized program 

References

  1. 1.
    Allender, E., Koucký, M.: Amplifying lower bounds by means of self-reducibility. J. ACM 57(3), 14:1–14:36 (2010). doi:10.1145/1706591.1706594 MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Appel, K., Haken, W.: Every planar map is four colorable. Part I. Discharging. Ill. J. Math. 21, 429–490 (1977a)MATHGoogle Scholar
  3. 3.
    Appel, K., Haken, W.: Every planar map is four colorable. Part II. Reducibility. Ill. J. Math. 21, 491–567 (1977b)MATHGoogle Scholar
  4. 4.
    Barish, R., Rothemund, P.W.K., Winfree, E.: Two computational primitives for algorithmic self-assembly: copying and counting. Nano Lett. 5(12), 2586–2592 (2005)CrossRefGoogle Scholar
  5. 5.
    Chow, T.Y.: Almost-natural proofs. J. Comput. Syst. Sci. 77(4), 728–737 (2011). doi:10.1016/j.jcss.2010.06.017 MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cook, M., Rothemund, P.W.K., Winfree, E.: Self-assembled circuit patterns. In: Procedings of 9th International Meeting on DNA Based Computers (DNA9), pp. 91–107. Springer, LNCS 2943 (2004)Google Scholar
  7. 7.
    Czeizler, E., Popa, A.: Synthesizing minimal tile sets for complex patterns in the framework of patterned DNA self-assembly. Theor. Comput. Sci. 499, 23–37 (2013)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Demaine, E.D., Demaine, M.L., Fekete, S.P., Patitz, M.J., Schweller, R.T., Winslow, A., Woods, D.: One tile to rule them all: simulating any Turing machine, tile assembly system, or tiling system with a single puzzle piece. Technical Report (2012). Arxiv preprint: arXiv:1212.4756
  9. 9.
    Demaine, E.D., Patitz, M.J., Rogers, T.A., Schweller, R.T., Summers, S.M., Woods, D.: The two-handed tile assembly model is not intrinsically universal. In: Proceedings of 40th International Colloquium on Automata, Languages and Programming (ICALP2013), Springer, Riga, Latvia, LNCS, vol. 7965, pp. 400–412 (2013). Arxiv preprint: arXiv:1306.6710
  10. 10.
    Doty, D., Lutz, J.H., Patitz, M.J., Summers, S.M., Woods, D.: Intrinsic universality in self-assembly. In: Proceedings of 27th International Symposium on Theoretical Aspects of Computer Science (STACS2009), pp. 275–286 (2009). Arxiv preprint: arXiv:1001.0208
  11. 11.
    Doty, D., Lutz, J.H., Patitz, M.J., Schweller, R.T., Summers, S.M., Woods, D.: The tile assembly model is intrinsically universal. In: Proceedings of 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS2012), pp. 439–446 (2012). Arxiv preprint: arXiv:1111.3097
  12. 12.
    Fujibayashi, K., Hariadi, R., Park, S.H., Winfree, E., Murata, S.: Toward reliable algorithmic self-assembly of DNA tiles: A fixed-width cellular automaton pattern. Nano Lett. 8(7), 1791–1797 (2007)CrossRefGoogle Scholar
  13. 13.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H, Freeman and Company (1979)Google Scholar
  14. 14.
    Gonthier, G.: Formal proof—the four-color theorem. Not. Am. Math. Soc. 55(11), 1382–1393 (2008)MathSciNetMATHGoogle Scholar
  15. 15.
    Göös, M., Lempiäinen, T., Czeizler, E., Orponen, P.: Search methods for tile sets in patterned DNA self-assembly. J. Comput. Syst. Sci. 80, 297–319 (2014)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Hales, T.C.: Cannonballs and honeycombs. Not. Am. Math. Soc. 47(4), 440–449 (2000)MathSciNetMATHGoogle Scholar
  17. 17.
    Helfgott, H.A.: The Ternary Goldbach Conjecture is True (2013). arXiv:1312.7748
  18. 18.
    Johnsen, A., Kao, M. Y., Seki, S.: Computing minimum tile sets to self-assemble color patterns. In: Proceedings of 24th International Symposium on Algorithms and Computation (ISAAC 2013), pp. 699–710. Springer, LNCS 8283 (2013)Google Scholar
  19. 19.
    Johnsen, A., Kao, M.Y., Seki, S.: A manually-checkable proof for the NP-hardness of 11-color pattern self-assembly tileset synthesis. J. Combin. Optim. (2015). In print. ArXiv preprint: arXiv:1409.1619
  20. 20.
    Kari, L., Kopecki, S., Meunier, P.E., Patitz, M.J., Seki, S.: Binary pattern tile set synthesis is NP-hard. In: Proceedings of 42nd International Colloquium on Automata, Languages, and Programming (ICALP2015), Springer, LNCS, vol. 9134, pp. 1022–1034 (2015a). Arxiv preprint: arXiv:1404.0967
  21. 21.
    Kari, L., Kopecki, S., Seki, S.: 3-Color bounded patterned self-assembly. Nat. Comp. 14(2), 279–292 (2015b)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Konev, B., Lisitsa, A.: A SAT attack on the Erdös discrepancy conjecture (2014). arXiv:1402.2184
  23. 23.
    Lathrop, J.I., Lutz, J.H., Patitz, M.J., Summers, S.M.: Computability and complexity in self-assembly. Theory Comput. Syst. 48(3), 617–647 (2011)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lund, K., Manzo, A.T., Dabby, N., Micholotti, N., Johnson-Buck, A., Nangreave, J., Taylor, S., Pei, R., Stojanovic, M.N., Walter, N.G., Winfree, E., Yan, H.: Molecular robots guided by prescriptive landscapes. Nature 465, 206–210 (2010)CrossRefGoogle Scholar
  25. 25.
    Ma, X., Lombardi, F.: Synthesis of tile sets for DNA self-assembly. IEEE Trans. Comput. Aid. D 27(5), 963–967 (2008)CrossRefGoogle Scholar
  26. 26.
    Marchal, C.: Study of the Kepler’s conjecture: the problem of the closest packing. Math. Z 267(3–4), 737–765 (2011)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Meunier, P.E., Patitz, M.J., Summers, S.M., Theyssier, G., Winslow, A., Woods, D.: Intrinsic universality in tile self-assembly requires cooperation. In: Proceedings of 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA2014), pp. 752–771 (2014). Arxiv preprint: arXiv:1304.1679
  28. 28.
    Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. J. ACM. 55(2):Article No. 11 (2008)Google Scholar
  29. 29.
    Patitz, M.J., Summers, S.M.: Self-assembly of decidable sets. Nat. Comput. 10(2), 853–877 (2011)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Qian, L., Winfree, E.: Scaling up digital circuit computation with DNA strand displacement cascades. Science 332(6034), 1196 (2011)CrossRefGoogle Scholar
  31. 31.
    Qian, L., Winfree, E., Bruck, J.: Neural network computation with DNA strand displacement cascades. Nature 475(7356), 368–372 (2011)CrossRefGoogle Scholar
  32. 32.
    Razborov, A.A., Rudich, S.: Natural proofs. In: Proceedings of 26th Annual ACM Symposium on Theory of Computing (STOC1994), pp. 204–213. ACM, New York (1994). doi:10.1145/195058.195134
  33. 33.
    Robertson, N., Sanders, D.P., Seymour, P., Thomas, R.: A new proof of the four-colour theorem. Electron. Res. Announc. AMS 2(1), 17–25 (1996)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Rothemund, P.W.: Folding DNA to create nanoscale shapes and patterns. Nature 440(7082), 297–302 (2006)CrossRefGoogle Scholar
  35. 35.
    Rothemund, P.W., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol. 2(12), 2041–2053 (2004)CrossRefGoogle Scholar
  36. 36.
    Rudich, S.: Super-bits, demi-bits, and NP/qpoly-natural proofs. J. Comput. Syst. Sci. 55, 204–213 (1997)MathSciNetGoogle Scholar
  37. 37.
    Seelig, G., Soloveichik, D., Zhang, D.Y., Winfree, E.: Enzyme-free nucleic acid logic circuits. Science 314(5805), 1585–1588 (2006)CrossRefGoogle Scholar
  38. 38.
    Seeman, N.C.: Nucleic-acid junctions and lattices. J. Theor. Biol. 99, 237–247 (1982)CrossRefGoogle Scholar
  39. 39.
    Seki, S.: Combinatorial optimization in pattern assembly (extended abstract). In: Proceedings of 12th International Conference on Unconventional Computation and Natural Computation (UCNC 2013), pp. 220–231. Springer, LNCS 7956 (2013)Google Scholar
  40. 40.
  41. 41.
    Szekeres, G., Peters, L.: Computer solution to the 17-point Erdös-Szekeres problem. ANZIAM J. 48, 151–164 (2006)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Tuckerman, B.: The 24th Mersenne prime. Proc. Natl. Acad. Sci. USA 68, 2319–2320 (1971)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Wang, H.: Proving theorems by pattern recognition-II. AT&T Tech. J. XL(1), 1–41 (1961)Google Scholar
  44. 44.
    Wei, B., Dai, M., Yin, P.: Complex shapes self-assembled from single-stranded DNA tiles. Nature 485(7400), 623–626 (2012)CrossRefGoogle Scholar
  45. 45.
    Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology (1998)Google Scholar
  46. 46.
    Winfree, E., Liu, F., Wenzler, L.A., Seeman, N.C.: Design and self-assembly of two-dimensional DNA crystals. Nature 394(6693), 539–44 (1998)CrossRefGoogle Scholar
  47. 47.
    Woods, D.: Intrinsic universality and the computational power of self-assembly (2013). Arxiv preprint: arXiv:1309.1265
  48. 48.
    Yan, H., Park, S.H., Finkelson, G., Reif, J.H., LaBean, T.H.: DNA-templated self-assembly of protein arrays and highly conductive nanowires. Science 301, 1882–1884 (2003)CrossRefGoogle Scholar
  49. 49.
    Yurke, B., Turberfield, A.J., Mills, A.P., Simmel, F.C., Neumann, J.L.: A DNA-fuelled molecular machine made of DNA. Nature 406(6796), 605–608 (2000)CrossRefGoogle Scholar
  50. 50.
    Zhang, J., Liu, Y., Ke, Y., Yan, H.: Periodic square-like gold nanoparticle arrays templated by self-assembled 2D DNA nanogrids on a surface. Nano Lett. 6(2), 248–251 (2006)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Lila Kari
    • 1
  • Steffen Kopecki
    • 1
  • Pierre-Étienne Meunier
    • 2
  • Matthew J. Patitz
    • 3
  • Shinnosuke Seki
    • 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.Department of Communication Engineering and InformaticsUniversity of Electro-CommunicationsTokyoJapan

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