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Maximizing Polynomials Subject to Assignment Constraints

  • Konstantin Makarychev
  • Maxim Sviridenko
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6755)

Abstract

We study the q-adic assignment problem. We first give an O(n (q − 1)/2)-approximation algorithm for the Koopmans–Beckman version of the problem improving upon the result of Barvinok. Then, we introduce a new family of instances satisfying “tensor triangle inequalities” and give a constant factor approximation algorithm for them. We show that many classical optimization problems can be modeled by q-adic assignment problems from this family. Finally, we give several integrality gap examples for the natural LP relaxations of the problem.

Keywords

Approximation Algorithm Assignment Problem Linear Programming Relaxation Performance Guarantee Quadratic Assignment Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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 16, 43–77 (1994)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Arkin, E., Hassin, R., Sviridenko, M.: Approximating the Maximum Quadratic Assignment Problem. Information Processing Letters 77, 13–16 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Babai, L., Codenotti, P.: Isomorphism of Hypergraphs of Low Rank in Moderately Exponential Time. In: Proc. 39th Ann. IEEE Symp. on Theory of Computing (FOCS 2008), pp. 667–676. IEEE Comp. Soc. Press, Los Alamitos (2008)Google Scholar
  4. 4.
    Barvinok, A.: Estimating L  ∞  norms by L 2k norms for functions on orbits. Found. Comput. Math. 2(4), 393–412 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bellare, M., Goldwasser, S., Lund, C., Russel, A.: Efficient Probabilistically Checkable Proofs and Applications to Approximation. In: Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing (STOC 1993), pp. 294–304 (1993)Google Scholar
  6. 6.
    Burkard, R.E., Cela, E.: Heuristics for biquadratic assignment problems and their computational comparison. European Journal of Operations Research 83, 283–300 (1995)CrossRefzbMATHGoogle Scholar
  7. 7.
    Burkard, R., Cela, E., Klinz, B.: On the biquadratic assignment problem. In: Quadratic Assignment and Related Problems, New Brunswick, NJ. DIMACS Ser. Discrete Math. Theoret. Comput. Sci., vol. 16, pp. 117–146. Amer. Math. Soc., Providence (1994)Google Scholar
  8. 8.
    Cela, E.: The Quadratic Assignment Problem: Theory and Algorithms. Springer, Heidelberg (1998)CrossRefzbMATHGoogle Scholar
  9. 9.
    Chor, B., Sudan, M.: A Geometric Approach to Betweenness. SIAM J. Discrete Math. 11(4), 511–523 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Guruswami, V., Håstad, J., Manokaran, R., Raghavendra, P., Charikar, M.: Beating the Random Ordering is Hard: Every ordering CSP is approximation resistant. Electronic Colloquium on Computational Complexity (ECCC) 18, 27 (2011)zbMATHGoogle Scholar
  11. 11.
    Hassin, R., Levin, A., Sviridenko, M.: Approximating the minimum quadratic assignment problems. ACM Transactions on Algorithms 6(1) (2009)Google Scholar
  12. 12.
    Koopmans, T.C., Beckman, M.: Assignment problems and the location of economic activities. Econometrica 25, 53–76 (1957)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lawler, E.: The quadratic assignment problem. Management Science 9, 586–599 (1962/1963)Google Scholar
  14. 14.
    Makarychev, Y.: Simple Linear Time Approximation Algorithm for Betweenness, Microsoft Research Technical Report MSR-TR-2009-74Google Scholar
  15. 15.
    Makarychev, K., Manokaran, R., Sviridenko, M.: Maximum Quadratic Assignment Problem: Reduction from Maximum Label Cover and LP-based Approximation Algorithm. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 594–604. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Nagarajan, V., Sviridenko, M.: On the maximum quadratic assignment problem. Mathematics of Operations Research 34(4), 859–868 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Queyranne, M.: Performance ratio of polynomial heuristics for triangle inequality quadratic assignment problems. Operations Research Letters 4, 231–234 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Winter, T., Zimmermann, U.: Real-time dispatch of trams in storage yards, Valparaiso. Mathematics of industrial systems, vol. IV (1996); Ann. Oper. Res. 96, 287–315 (2000)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Konstantin Makarychev
    • 1
  • Maxim Sviridenko
    • 1
  1. 1.IBM Thomas J. Watson Research CenterUSA

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