Advertisement

Solving NP-Complete Problems with Quantum Search

  • Martin Fürer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4957)

Abstract

In his seminal paper, Grover points out the prospect of faster solutions for an NP-complete problem like SAT. If there are n variables, then an obvious classical deterministic algorithm checks out all 2 n truth assignments in about 2 n steps, while his quantum search algorithm can find a satisfying truth assignment in about 2 n/2 steps.

For several NP-complete problems, many sophisticated classical algorithms have been designed. They are still exponential, but much faster than the brute force algorithms. The question arises whether their running time can still be decreased from T(n) to \(\tilde{O}(\sqrt{T(n)})\) by using a quantum computer. Isolated positive examples are known, and some speed-up has been obtained for wider classes. Here, we present a simple method to obtain the full T(n) to \(\tilde{O}(\sqrt{T(n)})\) speed-up for most of the many non-trivial exponential time algorithms for NP-hard problems. The method works whenever the widely used technique of recursive decomposition is employed.

This included all currently known algorithms for which such a speed-up has not yet been known.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Grover, L.K.: Quantum mechanics helps in searching for a needle in a haystack. Physical Review Letters 79(2), 325–328 (1997)CrossRefGoogle Scholar
  2. 2.
    Grover, L.K.: A framework for fast quantum mechanical algorithms. In: Proceedings of the 30th Annual ACM Symposium on Theory of Computing (STOC 1998), pp. 53–62. ACM Press, New York (1998)CrossRefGoogle Scholar
  3. 3.
    Angelsmark, O., Dahllöf, V., Jonsson, P.: Finite domain constraint satisfaction using quantum computation. In: Diks, K., Rytter, W. (eds.) MFCS 2002. LNCS, vol. 2420, pp. 93–103. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  4. 4.
    Eppstein, D.: Improved algorithms for 3-Coloring, 3-Edge-Coloring, and constraint satisfaction. In: Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), pp. 329–337. ACM Press, New York (2001)Google Scholar
  5. 5.
    Cerf, N., Grover, L., Williams, C.: Nested quantum search and structured problems. Phys. Rev. A 61(3) (2000) 14 032303.Google Scholar
  6. 6.
    Brassard, G., Høyer, P., Mosca, M., Tapp, A.: Quantum amplitude amplification and estimation. In: Lomonaco Jr., S.J., Brandt, H.E. (eds.) Quantum Computation and Information, AMS Contemporary Mathematics, vol. 305, pp. 53–74 (2002), http://arxiv.org/abs/quant-ph/0005055
  7. 7.
    Ambainis, A.: Quantum search algorithms. ACM SIGACT News 35(2), 22–35 (2004)CrossRefGoogle Scholar
  8. 8.
    Dantsin, E., Kreinovich, V., Wolpert, A.: On quantum versions of record-breaking algorithms for sat. ACM SIGACT News 36(4), 103–108 (2005)CrossRefGoogle Scholar
  9. 9.
    Schöning, U.: A probabilistic algorithm for k-SAT and constraint satisfaction problems. In: 40th Annual Symposium on Foundations of Computer Science (FOCS 1999), Washington - Brussels - Tokyo, pp. 410–414. IEEE, Los Alamitos (1999)Google Scholar
  10. 10.
    Davis, M., Putnam, H.: A computing procedure for quantification theory. Journal of Association Computer Machinery 7, 201–215 (1960)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Davis, M., Logemann, G., Loveland, D.: A machine program for theorem-proving. Communications of the ACM 5(7), 394–397 (1962)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Woeginger, G.: Exact algorithms for NP-hard problems: A survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  13. 13.
    Dantsin, E., Goerdt, A., Hirsch, E.A., Kannan, R., Kleinberg, J., Papadimitriou, C., Raghavan, P., Schöning, U.: A deterministic (2 − 2/(k + 1))n algorithm for k-SAT based on local search. Theoretical Computer Science 289(1), 69–83 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Kullmann, O.: New methods for 3-SAT decision and worst-case analysis. Theoretical Computer Science 223, 1–72 (1999)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Dahllöf, V., Jonsson, P., Wahlström, M.: Counting models for 2SAT and 3SAT formulae. Theoretical Computer Science 332(1-3), 265–291 (2005)CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortsch. Phys. 46, 493–506 (1998)CrossRefGoogle Scholar
  17. 17.
    Grover, L.K.: Quantum computers can search rapidly by using almost any transformation. Physical Review Letters 80, 4329–4332 (1998)CrossRefGoogle Scholar
  18. 18.
    Grover, L.K.: Rapid sampling through quantum computing. In: Proceedings of the 32nd Annual ACM Symposium on Theory of Computing (STOC 2000), pp. 618–626 (2000)Google Scholar
  19. 19.
    Chen, G., Sun, S.: Generalization of Grover’s algorithm to multiobject search in quantum computing, Part II: general unitary transformations. In: Brylinski, R., Chen, G. (eds.) Mathematics of Quantum Computation. Computational Mathematics, pp. 161–168. Chapman & Hall/CRC, Boca Raton, London, New York, Washington, D.C (2002)Google Scholar
  20. 20.
    Tarjan, R.E., Trojanowski, A.E.: Finding a maximum independent set. SIAM J. Comput. 6(3), 537–546 (1977)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Beigel, R.: Finding maximum independent sets in sparse and general graphs. In: SODA 1999. Proceedings of the tenth annual ACM-SIAM symposium on Discrete algorithms, Society for Industrial and Applied Mathematics, pp. 856–857 (1999)Google Scholar
  22. 22.
    Dantsin, E., Hirsch, E.A., Ivanov, S., Vsemirnov, M.: Algorithms for SAT and upper bounds on their complexity. Technical Report TR01-012, Electronic Colloquium on Computational Complexity (ECCC) (2001)Google Scholar
  23. 23.
    Fürer, M.: A faster algorithm for finding maximum independent sets in sparse graphs. In: Correa, J.R., Hevia, A., Kiwi, M. (eds.) LATIN 2006. LNCS, vol. 3887, pp. 491–501. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  24. 24.
    Fürer, M., Kasiviswanathan, S.P.: Exact Max 2-SAT: Easier and faster. In: van Leeuwen, J., Italiano, G.F., van der Hoek, W., Meinel, C., Sack, H., Plášil, F. (eds.) SOFSEM 2007. LNCS, vol. 4362, pp. 272–283. Springer, Heidelberg (2007)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Martin Fürer
    • 1
  1. 1.Department of Computer Science and EngineeringPennsylvania State UniversityUSA

Personalised recommendations