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A molecular solution for minimum vertex cover problem in tile assembly model

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Abstract

Self-assembly is a generalization of the crystal growth, which has been proposed as a mechanism for the bottom-up fabrication of autonomous DNA computation. In the same context, tile assembly model is a highly distributed parallel model of natural self-assembly. In this paper, we propose a tile assembly system to tackle a well-known NP-complete problem known as Minimum Vertex Cover problem. The proposed algorithm requires Θ(n×m) types of tiles, and each parallel assembly executes in a linear time, where n is the number of vertices and m is the number of edges. Furthermore, the experimental results proved the simplicity and the efficiency of the proposed algorithm to solve the Minimum Vertex Cover, and reduce the overall complexity to find the solution.

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References

  1. Feynman RP (1961) There’s plenty of room at the bottom. In: Gilbert HD (ed) Miniaturization. Reinhold, New York, pp 282–296

    Google Scholar 

  2. Head T, Rozenberg G, Bladergroen RB et al (2000) Computing with DNA by operating on plasmids. BioSystems 57:87–93

    Article  Google Scholar 

  3. Lipton RJ (1995) DNA solution of hard computational problem. Science 268(4):542–545

    Article  Google Scholar 

  4. Xu J, Tan G, Fan Y et al (2007) DNA computer principle, advances and difficulties (IV): on the models of DNA computer [J]. Chin J Comput 30(6):881–893

    MathSciNet  Google Scholar 

  5. Winfree E et al (1998) Design and self-assembly of two-dimensional DNA crystals. Nature 394:539–544

    Article  Google Scholar 

  6. Fu TJ, Seeman NC (1993) DNA double crossover molecules. Biochemistry 32:3211–3220

    Article  Google Scholar 

  7. Winfree E (1998) Algorithmic self-assembly of DNA [D]. California Institute of Technology, Pasadena, California

    Google Scholar 

  8. Reif JH (2002) Molecular assembly and computation: from theory to experimental demonstrations. In: Proceedings of the twenty-ninth international colloquium on automata, languages and programming, pp 1–21

    Chapter  Google Scholar 

  9. Robinson RM (1971) Undecidability and nonperiodicity for tilings of the plane. Invent Math 12(3):177–209

    Article  MathSciNet  MATH  Google Scholar 

  10. Wang Y, Hu P, Zhang X et al (2011) DNA self-assembly for the minimum vertex cover problem. Adv Sci Lett 4(1):74–79(6)

    Article  MathSciNet  Google Scholar 

  11. Wang H (1962) Dominoes and the AEA case of the decision problem. In: Proceedings of the symposium in the mathematical theory of automata, Brooklyn. Polytechnic Press, New York, pp 23–55

    Google Scholar 

  12. Chang W-L, Lin KWC, Chen J-C et al (2012) Molecular solutions of the RSA public-key cryptosystem on a DNA-based computer. J Supercomput. doi:10.1007/s11227-011-0627-z

    Google Scholar 

  13. Xiao D, Li W, Zhang Z, He L (2005) Solving maximum cut problems in the Adleman–Lipton model. BioSystems 82:203–207

    Article  Google Scholar 

  14. Dolati A, Haghighat MS, Safaei S, Mozaffar H (2008) Solving minimum beta-vertex separator problems in the Adleman–Lipton model. In: Proceedings of the 2008 international conference on foundations of computer science, FCS 2008, July 14–17, 2008, Las Vegas, Nevada, USA

    Google Scholar 

  15. Winfree E The Xgrow simulator. http://dna.caltech.edu/Xgrow/

  16. Garcia S, Orailoglu A (2009) Making DNA self-assembly error-proof: attaining small growth error rates through embedded information redundancy. In: Proceedings of DATE, pp 898–901

    Google Scholar 

  17. Adleman LM (1994) Molecular computation of solutions to combinatorial problems. Science 266(11):1021–1023

    Article  Google Scholar 

  18. Brun Y (2008) Solving NP-complete problems in the tile assembly model. Theor Comput Sci 395(1):31–46

    Article  MathSciNet  MATH  Google Scholar 

  19. Brun Y (2008) Arithmetic computation in the tile assembly model: addition and multiplication. Theor Comput Sci 378(1):17–31

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was partially funded by the Key Program of National Natural Science Foundation of China (Grant No. 61133005), and the National Natural Science Foundation of China (Grant Nos. 61070057, 61103047). Key Projects in the National Science & Technology Pillar Program (2012BAH09B02). The Ph.D. Programs Foundation of Ministry of Education of China (20100161110019). The project supported by the National Science Foundation for Distinguished Young Scholars of Hunan (12JJ1011). The Hunan Provincial Science and technology plan project (Grant No. 2012WK3053). The project supported by Scientific Research Fund of Hunan Provincial Education Department (Grant No.12A062). The Project of National Natural Science Foundation of China (Grant 61173013 and 61202109), The Project of the Office of Education in Zhejiang Province (Grant Y201226110).

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Authors and Affiliations

  1. College of Information Science and Technology, Hunan University, Changsha, 410082, China

    Fan Wu, Kenli Li, Ahmed Sallam & Xu Zhou

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  1. Fan Wu
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  2. Kenli Li
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  3. Ahmed Sallam
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  4. Xu Zhou
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Corresponding author

Correspondence to Kenli Li.

Appendix

Appendix

Several snapshots (Figs. 14, 15, 16 and 17) for applying the proposed algorithm using Xgrow simulator.

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A seed configuration with vertex set {1,3,4}

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Final configuration with vertex set {2,4,5}

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Final configuration with vertex set {1,3,4}

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Fig. 17
figure 18

Final configuration with vertex set {1,2,3,4}

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Wu, F., Li, K., Sallam, A. et al. A molecular solution for minimum vertex cover problem in tile assembly model. J Supercomput 66, 148–169 (2013). https://doi.org/10.1007/s11227-013-0892-0

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