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An Exact Algorithm for the Quadratic Assignment Problem on a Tree

Author

Listed:
  • Nicos Christofides

    (Imperial College, London, England)

  • Enrique Benavent

    (Universidad de Valencia, Valencia, Spain)

Abstract

The Tree QAP is a special case of the Quadratic Assignment Problem (QAP) where the nonzero flows form a tree. No condition is required for the distance matrix. This problem is NP-complete and is also a generalization of the Traveling Salesman Problem. In this paper, we present a branch-and-bound algorithm for the exact solution of the Tree QAP based on an integer programming formulation of the problem. The bounds are computed using a Lagrangian relaxation of this formulation. To solve the relaxed problem, we present a Dynamic Programming algorithm which is polynomially bounded. The obtained lower bound is very sharp and equals the optimum in many cases. This fact allows us to employ a reduction method to decrease the number of variables and leads to search-trees with a small number of nodes compared to those usually encountered in problems of this type. Computational results are given for problems with size up to 25.

Suggested Citation

  • Nicos Christofides & Enrique Benavent, 1989. "An Exact Algorithm for the Quadratic Assignment Problem on a Tree," Operations Research, INFORMS, vol. 37(5), pages 760-768, October.
    • Handle: RePEc:inm:oropre:v:37:y:1989:i:5:p:760-768
    DOI: 10.1287/opre.37.5.760
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    Cited by:

    1. Mikhail A. Bragin & Peter B. Luh & Joseph H. Yan & Nanpeng Yu & Gary A. Stern, 2015. "Convergence of the Surrogate Lagrangian Relaxation Method," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 173-201, January.
    2. Jingyang Zhou & Peter E.D. Love & Kok Lay Teo & Hanbin Luo, 2017. "An exact penalty function method for optimising QAP formulation in facility layout problem," International Journal of Production Research, Taylor & Francis Journals, vol. 55(10), pages 2913-2929, May.
    3. Ravindra K. Ahuja & Krishna C. Jha & James B. Orlin & Dushyant Sharma, 2007. "Very Large-Scale Neighborhood Search for the Quadratic Assignment Problem," INFORMS Journal on Computing, INFORMS, vol. 19(4), pages 646-657, November.
    4. José F. S. Bravo Ferreira & Yuehaw Khoo & Amit Singer, 2018. "Semidefinite programming approach for the quadratic assignment problem with a sparse graph," Computational Optimization and Applications, Springer, vol. 69(3), pages 677-712, April.

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