Maximum Quadratic Assignment Problem: Reduction from Maximum Label Cover and LP-Based Approximation Algorithm

  • Konstantin Makarychev
  • Rajsekar Manokaran
  • Maxim Sviridenko
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)

Abstract

We show that for every positive ε> 0, unless \({\mathcal NP} \subset {\mathcal BPQP}\), it is impossible to approximate the maximum quadratic assignment problem within a factor better than \(2^{\log^{1-\varepsilon} n}\) by a reduction from the maximum label cover problem. Then, we present an \(O(\sqrt{n})\)-approximation algorithm for the problem based on rounding of the linear programming relaxation often used in the state of the art exact algorithms.

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References

  1. 1.
    Adams, W.P., Johnson, T.A.: Improved Linear Programming-based Lower Bounds for the Quadratic Assignment Problem. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 16, pp. 43–77 (1994)Google Scholar
  2. 2.
    Anstreicher, K.: Recent advances in the solution of quadratic assignment problems. In: Anstreicher, K. (ed.) ISMP, 2003. Copenhagen. Math. Program, Ser. B, vol. 97(1-2), pp. 27–42 (2003)Google Scholar
  3. 3.
    Arkin, E., Hassin, R., Sviridenko, M.: Approximating the Maximum Quadratic Assignment Problem. Information Processing Letters 77, 13–16 (2001)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Arora, S., Frieze, A., Kaplan, H.: A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. Mathematical Programming 92(1), 1–36 (2002)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Arora, S., Lund, C.: Hardness of Approximations. In: Hochbaum, D. (ed.) Approximation Algorithms for NP-hard Problems. PWS Publishing (1996)Google Scholar
  6. 6.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. Journal of the ACM 45(3)Google Scholar
  7. 7.
    Arora, S., Safra, S.: Probabilistic checking of proofs: A new characterization of NP. Journal of the ACM 45(1), 70–122Google Scholar
  8. 8.
    Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting High Log-Densities – an \(O(n^{\frac{1}{4}})\) Approximation for Densest k-Subgraph. In: Proceedings of STOC (to appear, 2010)Google Scholar
  9. 9.
    Burkard, R.E., Cela, E., Pardalos, P., Pitsoulis, L.S.: The quadratic assignment problem. In: Du, D.Z., Pardalos, P.M. (eds.) Handbook of Combinatorial Optimization, vol. 3, pp. 241–339. Kluwer Academic Publishers, Dordrecht (1998)Google Scholar
  10. 10.
    Burkard, R.E., Dell’Amico, M., Martello, S.: Assignment Problems. SIAM, Philadelphia (2009)MATHGoogle Scholar
  11. 11.
    Cela, E.: The Quadratic Assignment Problem: Theory and Algorithms. Springer, Heidelberg (1998)MATHGoogle Scholar
  12. 12.
    Dong, Y., Wolkowicz, H.: A Low-Dimensional Semidefinite Relaxation for the Quadratic Assignment Problem. Mathematics of Operations Research 34, 1008–1022 (2009)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Dickey, J., Hopkins, J.: Campus building arrangement using TOPAZ. Transportation Science 6, 59–68 (1972)Google Scholar
  14. 14.
    Eiselt, H., Laporte, G.: A combinatorial optimization problem arising in dartboard design. Journal of Operational Research Society 42, 113–118 (1991)Google Scholar
  15. 15.
    Elshafei, A.: Hospital layout as a quadratic assignment problem. Operations Research Quarterly 28, 167–179 (1977)CrossRefGoogle Scholar
  16. 16.
    Feige, U.: Relations between average case complexity and approximation complexity. In: Proceedings of STOC 2002 , pp. 534–543 (2002)Google Scholar
  17. 17.
    Frieze, A., Kannan, R.: Quick approximation to matrices and applications. Combinatorica 19(2), 175–220 (1999)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Geoffrion, A., Graves, G.: Scheduling parallel production lines with changeover costs: Practical applications of a quadratic assignment/LP approach. Operations Research 24, 596–610 (1976)CrossRefGoogle Scholar
  19. 19.
    Hassin, R., Levin, A., Sviridenko, M.: Approximating the minimum quadratic assignment problems. ACM Transactions on Algorithms 6(1) (2009)Google Scholar
  20. 20.
    Khot, S.: Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique. In: Proceedings of FOCS 2004, pp. 136–145 (2004)Google Scholar
  21. 21.
    Koopmans, T.C., Beckman, M.: Assignment problems and the location of economic activities. Econometrica 25, 53–76 (1957)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Nagarajan, V., Sviridenko, M.: On the maximum quadratic assignment problem. Mathematics of Operations Research 34(4), 859–868 (2009); preliminary version appeared in Proceedings of SODA 2009, pp. 516–524Google Scholar
  23. 23.
    Laporte, G., Mercure, H.: Balancing hydraulic turbine runners: A quadratic assignment problem. European Journal of Operations Research 35, 378–381 (1988)CrossRefGoogle Scholar
  24. 24.
    Loilola, E.M., De Abreu, N.M.M., Boaventura-Netto, P.O., Hahn, P.M., Querido, T.: A survey for the quadratic assignment problem. Invited Review, European Journal of Operational Research 176, 657–690 (2006)CrossRefGoogle Scholar
  25. 25.
    Pardalos, P., Wolkowitz, H. (eds.): Proceedings of the DIMACS Workshop on Quadratic Assignment Problems. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 16 (1994)Google Scholar
  26. 26.
    Papadimitriou, C.H., Yannakakis, M.: The traveling salesman problem with distances one and two. Mathematics of Operations Research 18, 1–11 (1993)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Raz, R.: A Parallel Repetition Theorem. SIAM Journal on Computing 27, 763–803 (1998)MATHCrossRefMathSciNetGoogle Scholar
  28. 28.
    Queyranne, M.: Performance ratio of polynomial heuristics for triangle inequality quadratic assignment problems. Operations Research Letters 4, 231–234 (1986)MATHCrossRefMathSciNetGoogle Scholar
  29. 29.
    Steinberg, L.: The backboard wiring problem: a placement algorithm. SIAM Rev. 3, 37–50 (1961)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Zhao, Q., Karisch, S.E., Rendl, F., Wolkowicz, H.: Semidefinite Programming Relaxations for the Quadratic Assignment Problem. Journal of Combinatorial Optimization 2, 71–109 (1998)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Konstantin Makarychev
    • 1
  • Rajsekar Manokaran
    • 2
  • Maxim Sviridenko
    • 1
  1. 1.IBM Thomas J. Watson Research Center
  2. 2.Princeton UniversityPrinceton

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