Skip to main content
SpringerLink
Log in

Conic Characterization of Monge Matrices

  • Published:
Cybernetics and Systems Analysis Aims and scope

Abstract

A complete description is given to the linearity space of Monge cone matrices and of all its minimum faces. The description makes it possible to completely characterize Monge matrices. Possible applications of the results obtained are discussed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

REFERENCES

  1. E. L. Lawler, J. K. Lenstra, A. H. G. Rinnoy Kan, and D. B. Shmoys, The Traveling Salesman Problem, Wiley, Chichester (1985).

    Google Scholar 

  2. R. E. Burkard, V. G. Deineko, R. Van Dal et al., “Solvable cases of the traveling salesman problem: a survey,” SIAM Rev., 40, No. 3, 496–546 (1998).

    Google Scholar 

  3. R. E. Burkard, B. Klinz, and R. Rudolf, “Perspectives of Monge properties in optimization,” Discr. Appl. Math., 70, 95–161 (1996).

    Google Scholar 

  4. V. M. Demidenko and V. A. Shlyk, “On adjacency of polyhedrons vertices of combinatory optimization problems,” Dokl. AN BSSR, 33, No. 9, 40–44 (1989).

    Google Scholar 

  5. V. M. Demidenko and R. Rudolf, “A note on Kalmanson matrices,” Optimization, 40, 285–294 (1997).

    Google Scholar 

  6. V. M. Demidenko, “Conic characterization of some domains of strong solvability of the symmetric traveling salesman problem,” Dokl. NAN Belar., 43, No. 5, 17–21 (1999).

    Google Scholar 

  7. V. M. Demidenko, “Description of the cone of Kalmanson matrices in a minimum-dimension space,” Vestsi NAN BELARUSI, Ser. Fiz.-Mat. Navuk, No. 3, 1–6 (2000).

    Google Scholar 

  8. G. Monge, “Memoire sur la theorie des deblais et des remblais,” in: Histoire de l'Academie Royale des sciencas, Annee M. DCCLXXXI avec les Memoires de Mathematique et de Physique, pour la meme Annee, Tires des Registers de cette Academie, Paris (1781), pp. 666–704.

  9. D. A. Suprunenko, “Values of a linear form on a set of substitutions,” Kibernetika, No. 2, 59–63 (1968).

    Google Scholar 

  10. V. Ya. Burdyuk and V. N. Trofimov, “Generalization of the Gilmore Gomori results on solving the traveling salesman problem,” Izv. AN SSSR, Tekhn. Kibern., No. 3, 16–22 (1976).

    Google Scholar 

  11. V. G. Deineko and V. N. Trofimov, “Possible values of functions in special cases of the traveling salesman problem,” Kibernetika, No. 6, 135–136 (1981).

    Google Scholar 

  12. M. I. Rubinshtein, “Some properties of the problem on allocations,” Avtom. Telemekh., No. 1, 135–138 (1973).

    Google Scholar 

  13. D. A. Suprunenko, “Traveling salesman problem,” Kibernetika, No. 5, 121–124 (1975).

    Google Scholar 

  14. J. K. Park, “A special case of the n-vertex traveling salesman problem that can be solved in O n ()time,” Inform. Process. Lett., 40, 247–254 (1991).

    Google Scholar 

  15. R. E. Burkard and V. G. Deineko, “Polynomially solvable cases of the traveling salesman problem and a new exponential neighborhood,” Graz (Rep. Techn. Univ. Graz; Austria, No. A-8010) (1994).

  16. V. M. Demidenko, “A special case of the traveling salesman problem,” Vestsi AN BSSR, Ser. Fiz.-Mat. Navuk, No. 5, 28–32 (1976).

    Google Scholar 

  17. V. G. Deineko and O. F. Kalka, “Implementation of a combinatory algorithm,” in: Computer and Programming Actual Problems [in Russian], DGU, Dnepropetrovsk (1981), 40–47.

    Google Scholar 

  18. P. Brucker, “Monge-property and efficient algorithms,” in: Proc. Colloq. on Scheduling Theory and its Applications [in Russian], Minsk (1992).

  19. A. Hoffman, “On simple linear programming problems,” in: V. Klee (ed.), Convexity: Proc. 7th Symp. in Pure Math. of the AMS, Amer. Math. Soc., Providence, RI, 7, 317–327 (1963).

  20. A. Aggarwal, A. Bar-Noy, S. Khuller et al., “Efficient minimum cost matching using quadrangle inequality,” in: Proc. 33rd Ann. IEEE Symp. on Foundations of Computer Science, 583–593 (1992).

  21. R. Rudolf and G. J. Woeginger, “The cone of Monge matrices: extremal rays and applications,” ZOR, 42, 161–168 (1995).

    Google Scholar 

  22. S. N. Chernikov, Linear Inequalities [in Russian], Nauka, Moscow (1968).

    Google Scholar 

  23. A. Schrijver, Theory of Linear and Integer Programming, John Wiley and Sons, New York (1986).

    Google Scholar 

  24. V. G. Deineko, R. Rudolf, and G. J. Woeginger, “On the recognition of permuted Supnick and incomplete Monge matrices,” Acta Informatika, 33, 559–569 (1996).

    Google Scholar 

  25. F. Supnick, “Extreme Hamiltonian lines,” Ann. Math., 66, 179-201 (1957).

    Google Scholar 

  26. K. Kalmanson, “Edgeconvex circuits and the traveling salesman problem,” Can. J. Math., 27, 1000–1010 (1975).

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences of Belarus, Minsk, Belarus

    V. M. Demidenko

Authors
  1. V. M. Demidenko
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Demidenko, V.M. Conic Characterization of Monge Matrices. Cybernetics and Systems Analysis 40, 537–546 (2004). https://doi.org/10.1023/B:CASA.0000047875.57623.23

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:CASA.0000047875.57623.23

Search

Navigation