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Journal of Combinatorial Optimization

, Volume 31, Issue 3, pp 1269–1279 | Cite as

Linearizable special cases of the QAP

  • Eranda Çela
  • Vladimir G. Deineko
  • Gerhard J. Woeginger
Article

Abstract

We consider special cases of the quadratic assignment problem (QAP) that are linearizable in the sense of Bookhold. We provide combinatorial characterizations of the linearizable instances of the weighted feedback arc set QAP, and of the linearizable instances of the traveling salesman QAP. As a by-product, this yields a new well-solvable special case of the weighted feedback arc set problem.

Keywords

Combinatorial optimization Quadratic assignment problem  Linear assignment problem Computational complexity 

Notes

Acknowledgments

Part of this research was conducted while Vladimir Deineko and Gerhard Woeginger were visiting TU Graz, and they both thank the Austrian Science Fund (FWF): W1230, Doctoral Program in “Discrete Mathematics” for the financial support. Vladimir Deineko acknowledges support by Warwick University’s Centre for Discrete Mathematics and Its Applications (DIMAP). Gerhard Woeginger acknowledges support by DIAMANT (a mathematics cluster of the Netherlands Organization for Scientific Research NWO).

References

  1. Berenguer X (1979) A characterization of linear admissible transformations for the \(m\)-travelling salesmen problem. Eur J Oper Res 3:232–238MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bookhold I (1990) A contribution to quadratic assignment problems. Optimization 21:933–943MathSciNetCrossRefzbMATHGoogle Scholar
  3. Burkard RE, Çela E, Rote G, Woeginger GJ (1998) The quadratic assignment problem with a monotone anti-Monge and a symmetric Toeplitz matrix: easy and hard cases. Math Program B 82:125–158MathSciNetzbMATHGoogle Scholar
  4. Burkard RE, Deineko VG, van Dal R, van der Veen JAA, Woeginger GJ (1998) Well-solvable special cases of the TSP: a survey. SIAM Rev 40:496–546MathSciNetCrossRefzbMATHGoogle Scholar
  5. Burkard RE, Dell’Amico M, Martello S (2009) Assignment problems. SIAM, PhiladelphiaCrossRefzbMATHGoogle Scholar
  6. Çela E (1998) The quadratic assignment problem: theory and algorithms. Kluwer Academic Publishers, DordrechtCrossRefzbMATHGoogle Scholar
  7. Çela E, Deineko VG, Woeginger GJ (2012) Another well-solvable case of the QAP: maximizing the job completion time variance. Oper Res Lett 40:356–359MathSciNetCrossRefzbMATHGoogle Scholar
  8. Deineko VG, Woeginger GJ (1998) A solvable case of the quadratic assignment problem. Oper Res Lett 22:13–17MathSciNetCrossRefzbMATHGoogle Scholar
  9. Erdogan G (2006) Quadratic assignment problem: linearizations and polynomial time solvable cases. Ph.D. Thesis, Bilkent UniversityGoogle Scholar
  10. Erdogan G, Tansel BC (2007) A branch-and-cut algorithm for quadratic assignment problems based on linearizations. Comput Oper Res 34:1085–1106MathSciNetCrossRefzbMATHGoogle Scholar
  11. Erdogan G, Tansel BC (2011) Two classes of quadratic assignment problems that are solvable as linear assignment problems. Discret Opt 8:446–451MathSciNetCrossRefzbMATHGoogle Scholar
  12. Festa P, Pardalos PM, Resende MGC (2000) Feedback set problems. In: Du DZ, Pardalos PM (eds) Handbook of combinatorial optimization, supplement. Kluwer Academic Publishers, Philip Drive Norwell, pp 209–259Google Scholar
  13. Gabovich E (1976) Constant discrete programming problems on substitute sets (in Russian). Kibernetika (Kiev) 5:128–134 English tranlation inCybernetics 12:786–793 (1977)Google Scholar
  14. Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, San FranciscozbMATHGoogle Scholar
  15. Gilmore PC, Lawler EL, Shmoys DB (1985) Well-solved special cases. In: Lawler EL, Lenstra JK, Rinnooy Kan AHG, Shmoys DB (eds) The traveling salesman problem, vol 4. Wiley, Chichester, pp 87–143Google Scholar
  16. Kabadi SN, Punnen AP (2011) An \(O(n^4)\) algorithm for the QAP linearization problem. Math Oper Res 36:754–761MathSciNetCrossRefzbMATHGoogle Scholar
  17. Koopmans TC, Beckmann MJ (1957) Assignment problems and the location of economic activities. Econometrica 25:53–76MathSciNetCrossRefzbMATHGoogle Scholar
  18. Lawler EL, Lenstra JK, Rinnooy Kan AHG, Shmoys DB (1985) The traveling salesman problem. Wiley, ChichesterzbMATHGoogle Scholar
  19. Punnen AP, Kabadi SN (2013) A linear time algorithm for the Koopmans–Beckmann QAP linearization and related problems. Discret Opt 10:200–209MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Eranda Çela
    • 1
  • Vladimir G. Deineko
    • 2
  • Gerhard J. Woeginger
    • 3
  1. 1.Institut für Optimierung und Diskrete MathematikTU GrazGrazAustria
  2. 2.Warwick Business SchoolThe University of WarwickCoventryUK
  3. 3.Department of Mathematics and Computer ScienceTU EindhovenEindhovenThe Netherlands

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