Deterministic Algorithms for Multi-criteria TSP

  • Bodo Manthey
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6648)


We present deterministic approximation algorithms for the multi-criteria traveling salesman problem (TSP). Our algorithms are faster and simpler than the existing randomized algorithms.

First, we devise algorithms for the symmetric and asymmetric multi-criteria Max-TSP that achieve ratios of 1/2k − ε and 1/(4k − 2) − ε, respectively, where k is the number of objective functions. For two objective functions, we obtain ratios of 3/8 − ε and 1/4 − ε for the symmetric and asymmetric TSP, respectively. Our algorithms are self-contained and do not use existing approximation schemes as black boxes.

Second, we adapt the generic cycle cover algorithm for Min-TSP. It achieves ratios of 3/2 + ε, \(\frac 12 + \frac{\gamma^3}{1-3\gamma^2} + \varepsilon\), and \(\frac 12 + \frac{\gamma^2}{1-\gamma} + \varepsilon\) for multi-criteria Min-ATSP with distances 1 and 2, Min-ATSP with γ-triangle inequality and Min-STSP with γ-triangle inequality, respectively.


Approximation Algorithm Approximation Ratio Edge Weight Travel Salesman Problem Travel Salesman Problem 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Angel, E., Bampis, E., Gourvés, L.: Approximating the Pareto curve with local search for the bicriteria TSP(1,2) problem. Theoretical Computer Science 310(1-3), 135–146 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Angel, E., Bampis, E., Gourvès, L., Monnot, J. (Non)-Approximability for the Multi-criteria TSP(1,2). In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 329–340. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  3. 3.
    Arvind, V., Mukhopadhyay, P.: Derandomizing the Isolation Lemma and Lower Bounds for Circuit Size. In: Goel, A., Jansen, K., Rolim, J.D.P., Rubinfeld, R. (eds.) APPROX and RANDOM 2008. LNCS, vol. 5171, pp. 276–289. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Asadpour, A., Goemans, M.X., Madry, A., Gharan, S.O., Saberi, A.: An O(logn/loglogn)-approximation algorithm for the asymmetric traveling salesman problem. In: Proc. 21st Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 379–389. SIAM, Philadelphia (2010)CrossRefGoogle Scholar
  5. 5.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  6. 6.
    Berman, P., Karpinski, M.: 8/7-approximation algorithm for (1,2)-TSP. In: Proc. 17th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 641–648. SIAM, Philadelphia (2006)Google Scholar
  7. 7.
    Bläser, M.: A 3/4-approximation algorithm for maximum ATSP with weights zero and one. In: Jansen, K., Khanna, S., Rolim, J.D.P., Ron, D. (eds.) RANDOM 2004 and APPROX 2004. LNCS, vol. 3122, pp. 61–71. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Bläser, M., Manthey, B.: Approximating maximum weight cycle covers in directed graphs with weights zero and one. Algorithmica 42(2), 121–139 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bläser, M., Manthey, B., Sgall, J.: An improved approximation algorithm for the asymmetric TSP with strengthened triangle inequality. Journal of Discrete Algorithms 4(4), 623–632 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Böckenhauer, H.-J., Hromkovič, J., Klasing, R., Seibert, S., Unger, W.: Approximation algorithms for the TSP with sharpened triangle inequality. Information Processing Letters 75(3), 133–138 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Sunil Chandran, L., Shankar Ram, L.: On the relationship between ATSP and the cycle cover problem. Theoretical Computer Science 370(1-3), 218–228 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ehrgott, M.: Multicriteria Optimization. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  13. 13.
    Feige, U., Singh, M.: Improved Approximation Ratios for Traveling Salesperson Tours and Paths in Directed Graphs. In: Charikar, M., Jansen, K., Reingold, O., Rolim, J.D.P. (eds.) RANDOM 2007 and APPROX 2007. LNCS, vol. 4627, pp. 104–118. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Glaßer, C., Reitwießner, C., Witek, M.: Balanced combinations of solutions in multi-objective optimization, arXiv:1007.5475v1 [cs.DS] (2010)Google Scholar
  15. 15.
    Glaßer, C., Reitwießner, C., Witek, M.: Improved and derandomized approximations for two-criteria metric traveling salesman. Report 09-076, Rev. 1, Electron. Colloq. on Computational Complexity (ECCC) (2010)Google Scholar
  16. 16.
    Grandoni, F., Ravi, R., Singh, M.: Iterative Rounding for Multi-Objective Optimization Problems. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 95–106. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  17. 17.
    Kaplan, H., Lewenstein, M., Shafrir, N., Sviridenko, M.I.: Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs. Journal of the ACM 52(4), 602–626 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Manthey, B.: On approximating multi-criteria TSP. In: Proc. 26th Int. Symp. on Theoretical Aspects of Computer Science (STACS), pp. 637–648 (2009)Google Scholar
  19. 19.
    Manthey, B.: Multi-criteria TSP: Min and max combined. In: Bampis, E., Jansen, K. (eds.) WAOA 2009. LNCS, vol. 5893, pp. 205–216. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  20. 20.
    Manthey, B., Shankar Ram, L.: Approximation algorithms for multi-criteria traveling salesman problems. Algorithmica 53(1), 69–88 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Mulmuley, K., Vazirani, U.V., Vazirani, V.V.: Matching is as easy as matrix inversion. Combinatorica 7(1), 105–113 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Paluch, K., Mucha, M., Mądry, A.: A 7/9 - Approximation Algorithm for the Maximum Traveling Salesman Problem. In: Dinur, I., Jansen, K., Naor, J., Rolim, J. (eds.) APPROX 2009. LNCS, vol. 5687, pp. 298–311. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  23. 23.
    Papadimitriou, C.H., Yannakakis, M.: The complexity of restricted spanning tree problems. Journal of the ACM 29(2), 285–309 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: Proc. 41st Ann. IEEE Symp. on Foundations of Computer Science (FOCS), pp. 86–92. IEEE, Los Alamitos (2000)CrossRefGoogle Scholar
  25. 25.
    Zhang, T., Li, W., Li, J.: An improved approximation algorithm for the ATSP with parameterized triangle inequality. Journal of Algorithms 64(2-3), 74–78 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Bodo Manthey
    • 1
  1. 1.Department of Applied MathematicsUniversity of TwenteEnschedeThe Netherlands

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