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A comparison of approaches for finding minimum identifying codes on graphs

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Abstract

In order to formulate mathematical conjectures likely to be true, a number of base cases must be determined. However, many combinatorial problems are NP-hard and the computational complexity makes this research approach difficult using a standard brute force approach on a typical computer. One sample problem explored is that of finding a minimum identifying code. To work around the computational issues, a variety of methods are explored and consist of a parallel computing approach using MATLAB, an adiabatic quantum optimization approach using a D-Wave quantum annealing processor, and lastly using satisfiability modulo theory (SMT) and corresponding SMT solvers. Each of these methods requires the problem to be formulated in a unique manner. In this paper, we address the challenges of computing solutions to this NP-hard problem with respect to each of these methods.

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References

  1. Babbush, R., O’Gorman, B., Aspuru-Guzik, A.: Resource efficient gadgets for compiling adiabatic quantum optimization problems. arXiv:1307.8041v1 [quant-ph]

  2. Barrett, C., Sebastiani, R., Seshia, S.A., Tinelli, C.: Satisfiability modulo theories. In: Biere, A., Huele, M., van Maaren, H., Walsh, T. (eds.) Handbook of Satisfiability, pp. 825–885. Amsterdam, Netherlands, IOS Press (2009)

  3. Boixo, S., Albash, T., Spedalieri, F.M., Chancellor, N., Lidar, D.A.: Experimental signature of programmable quantum annealing. Nat. Commun. 4, 2067 (2013)

    Article  ADS  Google Scholar 

  4. Boutin, D., Horan, V., Pelto, M.: Identifying Codes on Directed De Bruijn Graphs. arXiv:1412.5842v2 (submitted)

  5. Cai, J., Macready, W.G., Roy, A.: A practical heuristic for finding graph minors. arXiv:1406.2741v1 [quant-ph]

  6. Charon, I., Hudry, O., Lobstein, A.: Minimizing the size of an identifying or locating-dominating code in a graph is NP-Hard. Theor. Comput. Sci. 290(3), 2109–2120 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Choi, V.: Minor-embedding in adiabatic quantum computation I. The parameter setting problem. Quantum Inf. Process. 7, 193–209 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Choi, V.: Minor-embedding in adiabatic quantum computation II. Minor-universal graph design. Quantum Inf. Process. 10, 343–353 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Farhi, E., Goldstone, J., Sipser, M.: Quantum Computation by Adiabatic Evolution. arXiv:quant-ph/0001106

  10. Gaitan, F., Clark, L.: Graph isomorphism and adiabatic quantum computing. Phys. Rev. A 89(2), 022342 (2014)

    Article  ADS  Google Scholar 

  11. Horan, V.: On the Existence of \(t\)-Identifying Codes in Undirected De Bruijn Graphs. arXiv:1508.00403

  12. Johnson, M.W., Amin, M.H.S., Gildert, S., Lanting, T., Hamze, F., Dickson, N., Harris, R., Berkley, A.J., Johansson, J., Bunyk, P., Chapple, E.M., Enderud, C., Hilton, J.P., Karimi, K., Ladizinsky, E., Ladizinsky, N., Oh, T., Perminov, I., Rich, C., Thom, M.C., Tolkacheva, E., Truncik, C.J.S., Uchaikin, S., Wang, J., Wilson, B., Rose, G.: Quantum annealing with manufactured spins. Nature 473, 194–198 (2011)

    Article  ADS  Google Scholar 

  13. Karpovsky, M.G., Chakrabarty, K., Levitin, L.B.: On a new class of codes for identifying vertices graphs. IEEE Trans. Inf. Theory 355(2), 599–611 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. King, A.D., McGeoch, C.C.: Algorithm engineering for a quantum annealing platform. arXiv:1410.2628v2 [cs.DS]

  15. Klymko, C., Sullivan, B., Humble, T.: Adiabatic quantum programming: minor embedding with hard faults. Quantum Inf. Process. 13, 709–729 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kreher, D.L., Stinson, D.R.: Combinatorial algorithms: generation, enumerations, and search. In: Discrete Mathematics and Its Applications, Book 7. CRC Press, Boca Raton (1998)

  17. Lucas, A.: Ising formulations of many NP problems. Front. Phys. 2, 5 (2014)

    Article  Google Scholar 

  18. Messiah, A.: Quantum Mechanics, vol. II. Wiley, New York (1976)

    MATH  Google Scholar 

  19. Mishra, V., Mathew, J., Pradhan, D.K.: Fault-tolerant de Bruijn graph base multipurpose architecture and routing protocol for wireless sensor networks. Int. J. Sens. Netw. 10(3), 160–175 (2011)

    Article  Google Scholar 

  20. Moussa, H., Baghdadi, A., Jezequel, M.: Binary de Briujn onchip network for a flexible multiprocessor LDPC decoder. ACM/IEEE Des. Autom. Conf. 429–434 (2008)

  21. Perdomo-Ortiz, A., Fluegemann, J., Biswas, R., Smelyanskiy, V.N.: A Performance Estimator for Quantum Annealers: Gauge Selection and Parameter Setting. arXiv:1503.01083v1 [quant-ph]

  22. Ray, S., Starobnski, D., Trachtenberg, A., Ungrangsi, R.: Robust location detection with sensor networks. IEEE J. Sel. Areas Commun. 22(6), 1016–1025 (2004)

    Article  Google Scholar 

  23. Rieffel, E.G., Venturelli, D., O’Gorman, B., Do, M.B., Prystay, E., Smelyanskiy, V.N.: A case study in programming a quantum annealer for hard operational planning problems. arXiv:1407.2887v1 [quant-ph]

  24. Xu, Y.C., Xiao, R.B.: Eighth ACIS International conference on software engineering, artificial intelligence, networking, and parallel/distributed computing, pp. 97–101 (2007)

  25. Zick, K.M., Shehab, O., French, M.: Experimental quantum annealing: case study involving the graph isomorphism problem. arXiv:1503.06453v1 [quant-ph]

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Acknowledgments

The work using the D-Wave quantum annealing machine was performed jointly by AFRL/RI and Lockheed Martin under Air Force Cooperative Research and Development Agreement 14-RI-CRADA-02. S. Adachi was supported by Internal Research and Development funding from Lockheed Martin. S. Adachi would also like to thank Todd Belote and Dr. Andy Dunn of Lockheed Martin for their assistance, respectively, with the SAT-to-Ising mapping in Sect. 2.2 and with the generation of models for the scaling analysis of larger cases in Sect. 2.5. LOCKHEED MARTIN and LOCKHEED are registered trademarks in the U.S. Patent and Trademark Office owned by Lockheed Martin Corporation.

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Authors and Affiliations

  1. Air Force Research Laboratory Information Directorate, Rome, NY, USA

    Victoria Horan & Stanley Bak

  2. Lockheed Martin, Palo Alto, CA, USA

    Steve Adachi

Authors
  1. Victoria Horan
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  2. Steve Adachi
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  3. Stanley Bak
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Correspondence to Victoria Horan.

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Approved for public release; distribution unlimited: 88ABW-2015-2163, DIS201511002.

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Horan, V., Adachi, S. & Bak, S. A comparison of approaches for finding minimum identifying codes on graphs. Quantum Inf Process 15, 1827–1848 (2016). https://doi.org/10.1007/s11128-016-1240-0

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  • DOI: https://doi.org/10.1007/s11128-016-1240-0

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