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Binary Pattern Tile Set Synthesis Is NP-Hard

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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 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.

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Notes

  1. This problem was claimed to be \(\mathbf {NP}\)-hard in a subsequent paper by the authors of [25] but what they proved was the \(\mathbf {NP}\)-hardness of a different problem (see [40]).

  2. http://self-assembly.net/wiki/index.php?title=2PATS-tileset-search (C++ version) and http://self-assembly.net/wiki/index.php?title=2PATS-search-ocaml (OCaml version).

  3. The implementation is Open MPI: http://www.open-mpi.org.

  4. https://www.sharcnet.ca/my/systems/show/41.

References

  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

    Article  MathSciNet  MATH  Google Scholar 

  2. Appel, K., Haken, W.: Every planar map is four colorable. Part I. Discharging. Ill. J. Math. 21, 429–490 (1977a)

    MATH  Google Scholar 

  3. Appel, K., Haken, W.: Every planar map is four colorable. Part II. Reducibility. Ill. J. Math. 21, 491–567 (1977b)

    MATH  Google Scholar 

  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)

    Article  Google Scholar 

  5. Chow, T.Y.: Almost-natural proofs. J. Comput. Syst. Sci. 77(4), 728–737 (2011). doi:10.1016/j.jcss.2010.06.017

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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. 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. 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. 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)

    Article  Google Scholar 

  13. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H, Freeman and Company (1979)

  14. Gonthier, G.: Formal proof—the four-color theorem. Not. Am. Math. Soc. 55(11), 1382–1393 (2008)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Hales, T.C.: Cannonballs and honeycombs. Not. Am. Math. Soc. 47(4), 440–449 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Helfgott, H.A.: The Ternary Goldbach Conjecture is True (2013). arXiv:1312.7748

  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)

  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. 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. Kari, L., Kopecki, S., Seki, S.: 3-Color bounded patterned self-assembly. Nat. Comp. 14(2), 279–292 (2015b)

    Article  MathSciNet  MATH  Google Scholar 

  22. Konev, B., Lisitsa, A.: A SAT attack on the Erdös discrepancy conjecture (2014). arXiv:1402.2184

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  25. Ma, X., Lombardi, F.: Synthesis of tile sets for DNA self-assembly. IEEE Trans. Comput. Aid. D 27(5), 963–967 (2008)

    Article  Google Scholar 

  26. Marchal, C.: Study of the Kepler’s conjecture: the problem of the closest packing. Math. Z 267(3–4), 737–765 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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. Mulzer, W., Rote, G.: Minimum-weight triangulation is NP-hard. J. ACM. 55(2):Article No. 11 (2008)

  29. Patitz, M.J., Summers, S.M.: Self-assembly of decidable sets. Nat. Comput. 10(2), 853–877 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  30. Qian, L., Winfree, E.: Scaling up digital circuit computation with DNA strand displacement cascades. Science 332(6034), 1196 (2011)

    Article  Google Scholar 

  31. Qian, L., Winfree, E., Bruck, J.: Neural network computation with DNA strand displacement cascades. Nature 475(7356), 368–372 (2011)

    Article  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  34. Rothemund, P.W.: Folding DNA to create nanoscale shapes and patterns. Nature 440(7082), 297–302 (2006)

    Article  Google Scholar 

  35. Rothemund, P.W., Papadakis, N., Winfree, E.: Algorithmic self-assembly of DNA Sierpinski triangles. PLoS Biol. 2(12), 2041–2053 (2004)

    Article  Google Scholar 

  36. Rudich, S.: Super-bits, demi-bits, and NP/qpoly-natural proofs. J. Comput. Syst. Sci. 55, 204–213 (1997)

    MathSciNet  Google Scholar 

  37. Seelig, G., Soloveichik, D., Zhang, D.Y., Winfree, E.: Enzyme-free nucleic acid logic circuits. Science 314(5805), 1585–1588 (2006)

    Article  Google Scholar 

  38. Seeman, N.C.: Nucleic-acid junctions and lattices. J. Theor. Biol. 99, 237–247 (1982)

    Article  Google Scholar 

  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)

  40. Sterling, A.: https://nanoexplanations.wordpress.com/2011/08/13/dna-self-assembly-of-multicolored-rectangles/ (2011)

  41. Szekeres, G., Peters, L.: Computer solution to the 17-point Erdös-Szekeres problem. ANZIAM J. 48, 151–164 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  42. Tuckerman, B.: The 24th Mersenne prime. Proc. Natl. Acad. Sci. USA 68, 2319–2320 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  43. Wang, H.: Proving theorems by pattern recognition-II. AT&T Tech. J. XL(1), 1–41 (1961)

    Google Scholar 

  44. Wei, B., Dai, M., Yin, P.: Complex shapes self-assembled from single-stranded DNA tiles. Nature 485(7400), 623–626 (2012)

    Article  Google Scholar 

  45. Winfree, E.: Algorithmic Self-Assembly of DNA. PhD thesis, California Institute of Technology (1998)

  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)

    Article  Google Scholar 

  47. Woods, D.: Intrinsic universality and the computational power of self-assembly (2013). Arxiv preprint: arXiv:1309.1265

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

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Acknowledgments

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. We also thank anonymous referees for their valuable comments, especially about the relationship between M-Sat and SetCover, on the earlier version of this paper.

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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

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

    Matthew J. Patitz

  4. Department of Communication Engineering and Informatics, University of Electro-Communications, 1-5-1, Chofugaoka, Chofu, Tokyo, 1828585, Japan

    Shinnosuke Seki

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  1. Lila Kari
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  5. Shinnosuke Seki
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Correspondence to Shinnosuke Seki.

Additional information

This is a full version of [20].

This work is supported in part by the NSERC Discovery Grant R2824A01 and UWO Faculty of Science grant to L. K., NSF Grant CCF-1219274 to P.-È. M., NSF Grants CCF-1117672 and CCF-1422152 to M. J. P., Academy of Finland, Postdoctoral Researcher Grant 13266670/T30606, JST Program to Disseminate Tenure Tracking System, MEXT, Japan 6F36, and JSPS Grant-in-Aid for Research Activity Start-up No. 15H06212 to S. S.

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Kari, L., Kopecki, S., Meunier, PÉ. et al. Binary Pattern Tile Set Synthesis Is NP-Hard. Algorithmica 78, 1–46 (2017). https://doi.org/10.1007/s00453-016-0154-7

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