Abstract
We consider traveling salesman problems (TSPs) with a permuted Monge matrix as cost matrix where the associated patching graph has a specially simple structure: a multistar, a multitree or a planar graph. In the case of multistars, we give a complete, concise and simplified presentation of Gaikov's theory. These results are then used for designing an O(m3 + mn) algorithm in the case of multitrees, where n is the number of cities and m is the number of subtours in an optimal assignment. Moreover we show that for planar patching graphs, the problem of finding an optimal subtour patching remains NP-complete.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
R.E. Burkard and V.G. Deineko, “Polynomially solvable cases of the traveling salesman problem and a new exponential neighborhood,” Computing, vol. 54, pp. 191–211, 1995.
R.E. Burkard, V.G. Deineko, R. van Dal, J.A.A. van der Veen, and G.J. Woeginger, “Well-solvable special cases of the TSP: A survey,” SIAM Reviews, vol. 40, pp. 496–546, 1998.
R.E. Burkard, B. Klinz, and R. Rudolf, “Perspectives of Monge properties in optimization,” Discrete Applied Mathematics, vol. 70, pp. 95–161, 1996.
V.Ya. Burdyuk and V.N. Trofimov, “Generalizations of the results of Gilmore and Gomory on the solution of the travelling salesman problem (in Russian),” Izv. Akad. Nauk SASS, Tech. Kibernet. vol. 3, pp. 16–22, 1976. English translation in Engineering Cybernetics, vol. 14, pp. 12–18, 1976.
V.G. Deineko, “Applying dynamic programming to solving a speical traveling salesman problem (in Russian),” Issledovanie operaziy i ASU Kiev, vol. 16, pp. 47–50, 1979.
V.G. Deineko and V.L. Filonenko, “On the reconstruction of specially structured matrices (in Russian),” Aktualnye Problemy EVMI Programmirovanie, Dnepropetrovsk, DGU, pp. 43–45, 1979.
H.N. Gabov, “Data structures for weighted matching and nearest common ancestors with linking,” in Proceedings of the First Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadelphia, pp. 434–443.
N.E. Gaikov, “On the minimization of a linear form on cycles (in Russian),” Vestsi Akad. Navuk BSSR Ser. Fiz.-Mat. Navuk, vol. 4, p. 128, 1980.
P.C. Gilmore and R.E. Gomory, “Sequencing a one state variable machine: a solvable case of the travelling salesman problem,” Oper. Research, vol. 12, pp. 655–679, 1964.
P.C. Gilmore, E.L. Lawler, and D.B. Shmoys, “Well-solved special cases,” chapter 4 in [12], pp. 87–143.
F. Glover and A.P. Punnen, “The traveling salesman problem: New solvable cases and linkages with the development of approximation algorithms,” Technical Report, University of Colorado, Boulder, 1995.
E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys, The Travelling Salesman Problem, Wiley, Chichester, 1985.
J.K. Park, “A special case of the n-vertex traveling salesman problem that can be solved in O(n) time,” Information Processing Letters, vol. 40, pp. 247–254, 1991.
J. Plesnik, “The NP-completeness of the Hamiltonian cycle problem in planar digraphs with degree bound two,” Information Processing Letters, vol. 8, pp. 199–201, 1979.
V.I. Sarvanov, “On the complexity of minimizing a linear form on a set of cyclic permutations (in Russian),” Dokl. Akad. Nauk SSSR, vol. 253, pp. 533–534. English translation in Soviet Math. Dokl., vol. 22, pp. 118–120, 1980.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Burkard, R.E., Deineko, V.G. & Woeginger, G.J. The Travelling Salesman Problem on Permuted Monge Matrices. Journal of Combinatorial Optimization 2, 333–350 (1998). https://doi.org/10.1023/A:1009768317347
Issue Date:
DOI: https://doi.org/10.1023/A:1009768317347