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On the Complexity of Graph Self-assembly in Accretive Systems

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DNA Computing (DNA 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4287))

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Abstract

We study the complexity of the Accretive Graph Assembly Problem (AGAP). An instance of AGAP consists of an edge-weighted graph G, a seed vertex in G, and a temperature τ. The goal is to determine if there is a sequence of vertex additions which constructs G starting from the seed. The edge weights model the forces of attraction and repulsion, and determine which vertices can be added to a partially assembled graph at the given temperature.

Our first result is that AGAP is NP-complete even on degree 3 planar graphs when edges have only two different types of weights. This resolves the complexity of AGAP in the sense that the problem is polytime solvable when either the degree is bounded by 2 or the number of distinct edge weights is one, and is NP-complete otherwise. Our second result is a dichotomy theorem that completely characterizes the complexity of AGAP on degree 3 bounded graphs with two distinct weights: w p , w n . We give a simple system of linear constraints on w p , w n , and τ that determines whether the problem is NP-complete or is polytime solvable. In the process of establishing this dichotomy, we give the first polytime algorithm to solve a non-trivial class of AGAP Finally, we consider the optimization version of AGAP where the goal is to realize a largest-possible subgraph of the given input graph. We show that even on constructible graphs of degree at most 3, it is NP-hard to realize a (1/n 1 − ε)-fraction of the input graph for any ε> 0; here n denotes the number of vertices in G.

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References

  1. Reif, J.H., Sahu, S., Yin, P.: Complexity of graph self-assembly in accretive systems and self-destructible systems. DNA Computing, 101–112 (2005)

    Google Scholar 

  2. Winfree, E., Liu, F., Wenzler, L.A., Seeman, N.C.: Design and self-assembly of two-dimensional DNA crystals. Nature 394, 539–544 (1998)

    Article  Google Scholar 

  3. Rothemund, P.: Using lateral capillary forces to compute by self-assembly. Proc. Nat. Acad. Sci. U.S.A. 97, 984–989 (2000)

    Article  Google Scholar 

  4. LaBean, T.H., Yan, H., Kopatsch, J., Liu, F., Winfree, E., Reif, J.H., Seeman, N.C.: Construction, analysis, ligation, and self-assembly of DNA triple crossover complexes. J. Amer. Chem. Soc. 122, 1848–1860 (2000)

    Article  Google Scholar 

  5. Yan, H., LaBean, T.H., Feng, L., Reif, J.H.: Directed nucleation assembly of DNA tile complexes for barcode-patterned lattices. Proc. Nat. Acad. Sci. U.S.A. 100, 8103–8108 (2003)

    Article  Google Scholar 

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

    Article  Google Scholar 

  7. Chelyapov, N., Brun, Y., Gopalkrishnan, M., Reishus, D., Shaw, B., Adleman, L.M.: DNA triangles and self-assembled hexagonal tilings. J. Amer. Chem. Soc. 126, 13924–13925 (2004)

    Article  Google Scholar 

  8. He, Y., Chen, Y., Liu, H., Ribbe, A.E., Mao, C.: Self-assembly of hexagonal DNA two-dimensional (2D) arrays. J. Amer. Chem. Soc. 127, 12202–12203 (2005)

    Article  Google Scholar 

  9. Malo, J., Mitchell, J.C., Vénien-Bryan, C., Harris, J.R., Wille, H., Sherratt, D.J., Turberfield, A.J.: Engineering a 2D protein-DNA crystal. Angewandte Chemie International Edition 44, 3057–3061 (2005)

    Article  Google Scholar 

  10. Wang, H.: Proving theorems by pattern recognition II. Bell Systems Technical Journal 40, 1–41 (1961)

    Google Scholar 

  11. Rothemund, P.W.K., Winfree, E.: The program-size complexity of self-assembled squares (extended abstract). In: STOC, pp. 459–468 (2000)

    Google Scholar 

  12. Winfree, E., Bekbolatov, R.: Proofreading tile sets: Error correction for algorithmic self-assembly. DNA Based Computers, 126–144 (2003)

    Google Scholar 

  13. Chen, H.L., Goel, A.: Error free self-assembly using error prone tiles. DNA Computing, 62–75 (2004)

    Google Scholar 

  14. Plesník, J.: The NP-completeness of the Hamiltonian cycle problem in planar digraphs with degree bound two. Inform. Process. Lett. 8, 199–201 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  15. Broersma, H., Li, X.: Spanning trees with many or few colors in edge-colored graphs. Discuss. Math. Graph Theory 17, 259–269 (1997)

    MATH  MathSciNet  Google Scholar 

  16. Adleman, L.M., Cheng, Q., Goel, A., Huang, M.D.A., Kempe, D., de Espanés, P.M., Rothemund, P.W.K.: Combinatorial optimization problems in self-assembly. In: STOC, pp. 23–32 (2002)

    Google Scholar 

  17. Adleman, L.M., Cheng, Q., Goel, A., Huang, M.D.A.: Running time and program size for self-assembled squares. In: STOC, pp. 740–748 (2001)

    Google Scholar 

  18. Aggarwal, G., Goldwasser, M., Kao, M.Y., Schweller, R.T.: Complexities for generalized models of self-assembly. In: SODA, pp. 880–889 (2004)

    Google Scholar 

  19. Sahu, S., Yin, P., Reif, J.H.: A self-assembly model of DNA tiles with time dependent glue strength. DNA Computing, 113–124 (2005)

    Google Scholar 

  20. Kao, M.Y., Schweller, R.: Reducing tile complexity for self-assembly through temperature programming. In: SODA, pp. 571–580 (2006)

    Google Scholar 

  21. Chen, H.L., Cheng, Q., Goel, A., Huang, M.D.A., de Espanés, P.M.: Invadable self-assembly: combining robustness with efficiency. In: SODA, pp. 890–899 (2004)

    Google Scholar 

  22. Fujibayashi, K., Murata, S.: A method of error suppression for self-assembling DNA tiles. DNA Computing, 113–127 (2004)

    Google Scholar 

  23. Reif, J.H., Sahu, S., Yin, P.: Compact error-resilient computational DNA tiling assemblies. DNA Computing, 293–307 (2004)

    Google Scholar 

  24. Schulman, R., Winfree, E.: Programmable control of nucleation for algorithmic self-assembly. DNA Computing, 319–328 (2004)

    Google Scholar 

  25. Soloveichik, D., Winfree, E.: Complexity of self-assembled shapes. DNA Computing, 344–354 (2004)

    Google Scholar 

  26. Soloveichik, D., Winfree, E.: Complexity of compact proofreading for self-assembled patterns. DNA Computing, 125–135 (2005)

    Google Scholar 

  27. Lagoudakis, M.G., LaBean, T.H.: 2D DNA self-assembly for satisfiability. DNA Based Computers, 139–152 (1999)

    Google Scholar 

  28. Cook, M., Rothemund, P.W.K., Winfree, E.: Self-assembled circuit patterns. DNA Based Computers, 91–107 (2003)

    Google Scholar 

  29. Schulman, R., Lee, S., Papadakis, N., Winfree, E.: One dimensional boundaries for DNA tile self-assembly. DNA Based Computers, 108–126 (2003)

    Google Scholar 

  30. Barish, R.D., Rothemund, P.W.K., Winfree, E.: Two computational primitives for algorithmic self-assembly: Copying and counting. Nano Letters 5, 2586–2592 (2005)

    Article  Google Scholar 

  31. Jonoska, N., Karl, S.A., Saito, M.: Three dimensional DNA structures in computing. BioSystems 52, 143–153 (1999)

    Article  Google Scholar 

  32. Jonoska, N., Sa-Ardyen, P., Seeman, N.C.: Computation by self-assembly of DNA graphs. Genetic Programming and Evolvable Machines 4, 123–137 (2003)

    Article  Google Scholar 

  33. Jonoska, N., McColm, G.L.: A Computational Model for Self-assembling Flexible Tiles. In: Calude, C.S., Dinneen, M.J., Păun, G., Jesús Pérez-Jímenez, M., Rozenberg, G. (eds.) UC 2005. LNCS, vol. 3699, pp. 142–156. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  34. Klavins, E., Ghrist, R., Lipsky, D.: A grammatical approach to self-organizing robotic systems. IEEE Trans. Automat. Control 51, 949–962 (2006)

    Article  MathSciNet  Google Scholar 

  35. Klavins, E.: Directed self-assembly using graph grammars. In: FNANO (2004)

    Google Scholar 

  36. Sa-Ardyen, P., Jonoska, N., Seeman, N.C.: Self-assembling DNA graphs. Natural Computing 2, 427–438 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11, 329–343 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  38. Middleton, A.A.: Computational complexity of determining the barriers to interface motion in random systems. Phys. Rev. E 59, 2571–2577 (1999)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Computer and Information Science, School of Engineering and Applied Sciences, University of Pennsylvania, Philadelphia, PA, 19104, USA

    Stanislav Angelov, Sanjeev Khanna & Mirkó Visontai

Authors
  1. Stanislav Angelov
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  2. Sanjeev Khanna
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  3. Mirkó Visontai
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Editors and Affiliations

  1. Department of Chemistry, Purdue University, 560 Oval Drive, 47907-2084, West Lafayette, IN, USA

    Chengde Mao

  2. Faculty of Education and Integrated Arts and Sciences, Department of Mathematics, Waseda University, 1-6-1 Nishiwaseda, Shinjuku-ku, 169-8050, Tokyo, Japan

    Takashi Yokomori

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© 2006 Springer-Verlag Berlin Heidelberg

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Angelov, S., Khanna, S., Visontai, M. (2006). On the Complexity of Graph Self-assembly in Accretive Systems. In: Mao, C., Yokomori, T. (eds) DNA Computing. DNA 2006. Lecture Notes in Computer Science, vol 4287. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11925903_8

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  • DOI: https://doi.org/10.1007/11925903_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-49024-1

  • Online ISBN: 978-3-540-68423-7

  • eBook Packages: Computer ScienceComputer Science (R0)

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