Algorithmica

, Volume 53, Issue 1, pp 69–88 | Cite as

Approximation Algorithms for Multi-Criteria Traveling Salesman Problems

Article

Abstract

We analyze approximation algorithms for several variants of the traveling salesman problem with multiple objective functions. First, we consider the symmetric TSP (STSP) with γ-triangle inequality. For this problem, we present a deterministic polynomial-time algorithm that achieves an approximation ratio of \(\min\{1+\gamma,\frac{2\gamma^{2}}{2\gamma^{2}-2\gamma +1}\}+\varepsilon\) and a randomized approximation algorithm that achieves a ratio of \(\frac{2\gamma^{3}+2\gamma^{2}}{3\gamma^{2}-2\gamma +1}+\varepsilon\) . In particular, we obtain a 2+ε approximation for multi-criteria metric STSP.

Then we show that multi-criteria cycle cover problems admit fully polynomial-time randomized approximation schemes. Based on these schemes, we present randomized approximation algorithms for STSP with γ-triangle inequality (ratio \(\frac{1+\gamma}{1+3\gamma -4\gamma^{2}}+\varepsilon\) ), asymmetric TSP (ATSP) with γ-triangle inequality (ratio \(\frac{1}{2}+ \frac{\gamma^{3}}{1-3\gamma^{2}}+\varepsilon\) ), STSP with weights one and two (ratio 4/3) and ATSP with weights one and two (ratio 3/2).

Keywords

Approximation algorithms Multi-criteria optimization Traveling salesman problem 

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References

  1. 1.
    Angel, E., Bampis, E., Gourvés, L.: Approximating the Pareto curve with local search for the bicriteria TSP(1,2) problem. Theor. Comput. Sci. 310(1–3), 135–146 (2004) MATHCrossRefGoogle Scholar
  2. 2.
    Angel, E., Bampis, E., Gourvés, L., Monnot, J.: (Non-)approximability for the multi-criteria TSP(1,2). In: Liśkiewicz, M., Reischuk, R. (eds.) Proc. of the 15th Int. Symp. on Fundamentals of Computation Theory (FCT). Lecture Notes in Computer Science, vol. 3623, pp. 329–340. Springer, New York (2005) CrossRefGoogle Scholar
  3. 3.
    Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M.: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, New York (1999) MATHGoogle Scholar
  4. 4.
    Barahona, F., Pulleyblank, W.R.: Exact arborescences, matchings and cycles. Discret. Appl. Math. 16(2), 91–99 (1987) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Berman, P., Karpinski, M.: 8/7-approximation algorithm for (1,2)-TSP. In: Proc. of the 17th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), pp. 641–648. SIAM, Philadelphia (2006) CrossRefGoogle Scholar
  6. 6.
    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.) Proc. of the 7th Int. Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX). Lecture Notes in Computer Science, vol. 3122, pp. 61–71. Springer, New York (2004) Google Scholar
  7. 7.
    Bläser, M., Manthey, B., Sgall, J.: An improved approximation algorithm for the asymmetric TSP with strengthened triangle inequality. J. Discret. Algorithms 4(4), 623–632 (2006) MATHCrossRefGoogle Scholar
  8. 8.
    Böckenhauer, H.-J., Hromkovič, J., Klasing, R., Seibert, S., Unger, W.: Approximation algorithms for the TSP with sharpened triangle inequality. Inf. Process. Lett. 75(3), 133–138 (2000) CrossRefGoogle Scholar
  9. 9.
    Chandran, L.S., Ram, L.S.: On the relationship between ATSP and the cycle cover problem. Theor. Comput. Sci. 370(1-3), 218–228 (2007) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Christofides, N.: Worst-case analysis of a new heuristic for the traveling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, Pennsylvania, USA (1976) Google Scholar
  11. 11.
    Ehrgott, M.: Approximation algorithms for combinatorial multicriteria optimization problems. Int. Trans. Oper. Res. 7(1), 5–31 (2000) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Ehrgott, M.: Multicriteria Optimization. Springer, New York (2005) MATHGoogle Scholar
  13. 13.
    Ehrgott, M., Gandibleux, X.: A survey and annotated bibliography of multiobjective combinatorial optimization. OR Spectrum 22(4), 425–460 (2000) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York (1979) MATHGoogle Scholar
  15. 15.
    Gilmore, P.C., Lawler, E.L., Shmoys, D.B.: Well-solved special cases. In: Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.) The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, pp. 87–143. Wiley, New York (1985) Google Scholar
  16. 16.
    Gutin, G., Punnen, A.P. (eds.): The Traveling Salesman Problem and its Variations. Kluwer Academic, Dordrecht (2002) MATHGoogle Scholar
  17. 17.
    Kaplan, H., Lewenstein, M., Shafrir, N., Sviridenko, M.I.: Approximation algorithms for asymmetric TSP by decomposing directed regular multigraphs. J. ACM 52(4), 602–626 (2005) CrossRefMathSciNetGoogle Scholar
  18. 18.
    Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.): The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. Wiley, New York (1985) MATHGoogle Scholar
  19. 19.
    Lovász, L., Plummer, M.D.: Matching Theory. North-Holland Mathematics Studies, vol. 121. Elsevier, Amsterdam (1986) MATHGoogle Scholar
  20. 20.
    Mulmuley, K., Vazirani, U.V., Vazirani, V.V.: Matching is as easy as matrix inversion. Combinatorica 7(1), 105–113 (1987) MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Papadimitriou, C.H.: The complexity of restricted spanning tree problems. J. ACM 29(2), 285–309 (1982) MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: Proc. of the 41st Ann. IEEE Symp. on Foundations of Computer Science (FOCS), pp. 86–92. IEEE Computer Society (2000) Google Scholar
  23. 23.
    Rosenkrantz, D.J., Stearns, R.E., Lewis II, P.M.: An analysis of several heuristics for the traveling salesman problem. SIAM J. Comput. 6(3), 563–581 (1977) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Tutte, W.T.: A short proof of the factor theorem for finite graphs. Can. J. Math. 6, 347–352 (1954) MATHMathSciNetGoogle Scholar
  25. 25.
    Vazirani, V.V.: Approximation Algorithms. Springer, New York (2001) Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Department of Computer ScienceYale UniversityNew HavenUSA
  2. 2.Institut für Theoretische InformatikETH ZürichZürichSwitzerland

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