skip to main content
research-article

Approximating the minimum quadratic assignment problems

Published: 28 December 2009 Publication History

Abstract

We consider the well-known minimum quadratic assignment problem. In this problem we are given two n × n nonnegative symmetric matrices A = (aij) and B = (bij). The objective is to compute a permutation π of V = {1,…,n} so that ∑ i,jVij aπ(i),π(j)bi,j is minimized.
We assume that A is a 0/1 incidence matrix of a graph, and that B satisfies the triangle inequality. We analyze the approximability of this class of problems by providing polynomial bounded approximations for some special cases, and inapproximability results for other cases.

References

[1]
Altinkemer, K., and Gavish, B. 1988. Heuristics with constant error guarantees for the design of tree networks. Manag. Sci. 34, 331--341.
[2]
Anstreicher, K. 2003. Recent advances in the solution of quadratic assignment problems. Math. Program. Ser. B 97, 1--2, 27--42.
[3]
Arkin, E., Chiang, Y.-J., Mitchell, J. S. B., Skiena, S., and Yang, T.-C. 1999. On the maximum scatter traveling salesperson problem. SIAM J. Comput. 29, 515--544.
[4]
Arkin, E., Hassin, R., and Sviridenko, M. 2001. Approximating the maximum quadratic assignment problem. Inform. Process. Lett. 77, 13--16.
[5]
Arora, S., Frieze, A., and Kaplan, H. 2002. A new rounding procedure for the assignment problem with applications to dense graph arrangement problems. Math. Program. Ser. A 92, 1, 1--36.
[6]
Cela, E. 1998. The Quadratic Assignment Problem: Theory and Algorithms. Kluwer Academic, Dordrecht, The Netherlands.
[7]
Charikar, M., Hajiaghayi, M., and Karloff, H. 2006. Satish rao: 22 spreading metrics for vertex ordering problems. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'06). 1018--1027.
[8]
Christofides, N. 1976. Worst-Case analysis of a new heuristic for the traveling salesman problem. Tech. rep. 338, Graduate School of Industrial Administration, Carnegie Mellon University.
[9]
Dickey, J., and Hopkins, J. 1972. Campus building arrangement using TOPAZ. Transport. Sci. 6, 59--68.
[10]
Eiselt, H., and Laporte, G. 1991. A combinatorial optimization problem arising in dartboard design. J. Oper. Res. Soc. 42, 113--118.
[11]
Elshafei, A. 1977. Hospital layout as a quadratic assignment problem. Oper. Res. Quart. 28, 167--179.
[12]
Feige, U., and Lee, J. 2007. An improved approximation ratio for the minimum linear arrangement problem. Inform. Process. Lett. 101, 26--29.
[13]
Garey, M. R., and Johnson, D. S. 1979. Computers and Intractability A Guide to the Theory of NP-Completeness. W. H. Freeman, New York.
[14]
Garg, N. 2005. Saving an epsilon: A 2-approximation for the k-MST problem in graphs. In Proceedings of the Annual ACM Symposium on Theory of Computing (STOC'05). 396--402.
[15]
Geoffrion, A., and Graves, G. 1976. Scheduling parallel production lines with changeover costs: Practical applications of a quadratic assignment/LP approach. Oper. Res. 24, 596--610.
[16]
Goemans, M. X., and Williamson, D. P. 1995. A general approximation technique for constrained forest problems. SIAM J. Comput. 24, 296--317.
[17]
Guttmann-Beck, N., and Hassin, R. 1998a. Approximation algorithms for minimum tree partition. Discr. Appl. Math. 87, 117--137.
[18]
Guttmann-Beck, N., and Hassin, R. 1998b. Approximation algorithms for min-sum p-clustering. Discr. Appl. Math. 89, 125--142.
[19]
Hoogeveen, J. A. 1991. Analysis of Christofides' heuristic: Some paths are more difficult than cycles. Oper. Res. Lett. 10, 291--295.
[20]
Laporte, G., and Mercure, H. 1988. Balancing hydraulic turbine runners: A quadratic assignment problem. Eur. J. Oper. Res. 35, 378--381.
[21]
Nagarajan, V., and Sviridenko, M. 2009. On the maximum quadratic assignment problem. to appear in Math. Oper. Res. Preliminary version appeared in Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'09). 516--524.
[22]
Queyranne, M. 1986. Performance ratio of polynomial heuristics for triangle inequality quadratic assignment problems. Oper. Res. Lett. 4, 231--234.
[23]
Sahni, S., and Gonzalez, T. 1976. P-Complete approximation problems. J. ACM 23, 555--565.
[24]
Steinberg, L. 1961. The backboard wiring problem: A placement algorithm. SIAM Rev. 3, 37--50.

Cited By

View all
  • (2023)Qubit Allocation for Distributed Quantum ComputingIEEE INFOCOM 2023 - IEEE Conference on Computer Communications10.1109/INFOCOM53939.2023.10228915(1-10)Online publication date: 17-May-2023
  • (2020)A variable neighbourhood search enhanced estimation of distribution algorithm for quadratic assignment problemsOPSEARCH10.1007/s12597-020-00475-4Online publication date: 28-Aug-2020
  • (2017)Maximizing Polynomials Subject to Assignment ConstraintsACM Transactions on Algorithms10.1145/314713713:4(1-15)Online publication date: 14-Nov-2017
  • Show More Cited By

Index Terms

  1. Approximating the minimum quadratic assignment problems

    Recommendations

    Comments

    Information & Contributors

    Information

    Published In

    cover image ACM Transactions on Algorithms
    ACM Transactions on Algorithms  Volume 6, Issue 1
    December 2009
    374 pages
    ISSN:1549-6325
    EISSN:1549-6333
    DOI:10.1145/1644015
    Issue’s Table of Contents
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

    Publisher

    Association for Computing Machinery

    New York, NY, United States

    Publication History

    Published: 28 December 2009
    Accepted: 01 September 2009
    Revised: 01 March 2009
    Received: 01 September 2006
    Published in TALG Volume 6, Issue 1

    Permissions

    Request permissions for this article.

    Check for updates

    Author Tags

    1. Approximation algorithms
    2. quadratic assignment problem

    Qualifiers

    • Research-article
    • Research
    • Refereed

    Contributors

    Other Metrics

    Bibliometrics & Citations

    Bibliometrics

    Article Metrics

    • Downloads (Last 12 months)13
    • Downloads (Last 6 weeks)6
    Reflects downloads up to 20 Jan 2025

    Other Metrics

    Citations

    Cited By

    View all
    • (2023)Qubit Allocation for Distributed Quantum ComputingIEEE INFOCOM 2023 - IEEE Conference on Computer Communications10.1109/INFOCOM53939.2023.10228915(1-10)Online publication date: 17-May-2023
    • (2020)A variable neighbourhood search enhanced estimation of distribution algorithm for quadratic assignment problemsOPSEARCH10.1007/s12597-020-00475-4Online publication date: 28-Aug-2020
    • (2017)Maximizing Polynomials Subject to Assignment ConstraintsACM Transactions on Algorithms10.1145/314713713:4(1-15)Online publication date: 14-Nov-2017
    • (2017)Online Placement of Multi-Component Applications in Edge Computing EnvironmentsIEEE Access10.1109/ACCESS.2017.26659715(2514-2533)Online publication date: 2017
    • (2015)Minimum Congestion Mapping in a CloudSIAM Journal on Computing10.1137/11084523944:3(819-843)Online publication date: Jan-2015
    • (2015)Memetic search for the quadratic assignment problemExpert Systems with Applications: An International Journal10.1016/j.eswa.2014.08.01142:1(584-595)Online publication date: 1-Jan-2015
    • (2014)Maximum Quadratic Assignment ProblemACM Transactions on Algorithms10.1145/262967210:4(1-18)Online publication date: 13-Aug-2014
    • (2013)Automating Cloud Network Optimization and EvolutionIEEE Journal on Selected Areas in Communications10.1109/JSAC.2013.13120431:12(2620-2631)Online publication date: Dec-2013
    • (2012)NetDEOProceedings of the 2012 IEEE 20th International Workshop on Quality of Service10.5555/2330748.2330778(1-9)Online publication date: 4-Jun-2012
    • (2012)Dynamic programming for the quadratic assignment problem on treesAutomation and Remote Control10.1134/S000511791202011773:2(336-348)Online publication date: 1-Feb-2012
    • Show More Cited By

    View Options

    Login options

    Full Access

    View options

    PDF

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader

    Media

    Figures

    Other

    Tables

    Share

    Share

    Share this Publication link

    Share on social media