(Non)-Approximability for the Multi-criteria TSP(1,2)

  • Eric Angel
  • Evripidis Bampis
  • Laurent Gourvès
  • Jérôme Monnot
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3623)


Many papers deal with the approximability of multi-criteria optimization problems but only a small number of non-approximability results, which rely on NP-hardness, exist in the literature. In this paper, we provide a new way of proving non-approximability results which relies on the existence of a small size good approximating set (i.e. it holds even in the unlikely event of P=NP). This method may be used for several problems but here we illustrate it for a multi-criteria version of the traveling salesman problem with distances one and two (TSP(1,2)). Following the article of Angel et al. (FCT 2003) who presented an approximation algorithm for the bi-criteria TSP(1,2), we extend and improve the result to any number k of criteria.


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  1. 1.
    Angel, E., Bampis, E., Gourvés, L.: Approximating the Pareto curve with local search for the bi-criteria TSP(1,2) problem. In: Lingas, A., Nilsson, B.J. (eds.) FCT 2003. LNCS, vol. 2751, pp. 39–48. Springer, Heidelberg (2003)Google Scholar
  2. 2.
    Angel, E., Bampis, E., Gourvès, L., Monnot, J.: (Non)-Approximability for the multi-criteria TSP(1, 2). Technical Report, Université d’Évry Val d’Essonne, No 116-2005 (2005)Google Scholar
  3. 3.
    Deng, X., Papadimitriou, C.H., Safra, S.: On the Complexity of Equilibria. In: Proc. of STOC 2002, pp. 67–71 (2002)Google Scholar
  4. 4.
    Engebretsen, L.: An Explicit Lower Bound for TSP with Distances One and Two. Algorithmica 35(4), 301–318 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Fotakis, D., Spirakis, P.: A Hamiltonian Approach to the Assignment of Non- Reusable Frequencies. In: Arvind, V., Sarukkai, S. (eds.) FST TCS 1998. LNCS, vol. 1530, pp. 18–29. Springer, Heidelberg (1998)Google Scholar
  6. 6.
    Johnson, D.S., Papadimitriou, C.H.: Performance guarantees for heuristics. In: Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B., Lawler, E.L. (eds.) The Traveling Salesman Problem: a guided tour of Combinatorial Optimization, pp. 145–180. Wiley Chichester, Chichester (1985)Google Scholar
  7. 7.
    Lucas, D.E.: Récréations mathématiques. vol. II. Gauthier Villars, Paris (1892)Google Scholar
  8. 8.
    Papadimitriou, C.H., Yannakakis, M.: On the approximability of trade-offs and optimal access of web sources. In: Proc. of FOCS 2000, pp. 86–92 (2000)Google Scholar
  9. 9.
    Papadimitriou, C.H., Yannakakis, M.: The traveling salesman problem with distances one and two. Mathematics of Operations Research 18(1), 1–11 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Rosenkrantz, D.J., Stearns, R.E., Lewis II, P.M.: An analysis of several heuristics for the traveling salesman problem. SIAM J. Comp. 6, 563–581 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Warburton, A.: Approximation of Pareto optima in multiple-objective shortest path problems. Operations Research 35(1), 70–79 (1987)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Eric Angel
    • 1
  • Evripidis Bampis
    • 1
  • Laurent Gourvès
    • 1
  • Jérôme Monnot
    • 2
  1. 1.LaMI, CNRS UMR 8042Université d’Évry Val d’EssonneFrance
  2. 2.LAMSADE, CNRS UMR 7024Université de Paris-DauphineFrance

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