Towards the Notion of Stability of Approximation for Hard Optimization Tasks and the Traveling Salesman Problem

(Extended Abstract)
  • Hans-Joachim Böckenhauer
  • Juraj Hromkovič
  • Ralf Klasing
  • Sebastian Seibert
  • Walter Unger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1767)


The investigation of the possibility to efficiently compute approximations of hard optimization problems is one of the central and most fruitful areas of current algorithm and complexity theory. The aim of this paper is twofold. First, we introduce the notion of stability of approximation algorithms. This notion is shown to be of practical as well as of theoretical importance, especially for the real understanding of the applicability of approximation algorithms and for the determination of the border between easy instances and hard instances of optimization problems that do not admit any polynomial-time approximation.

Secondly, we apply our concept to the study of the traveling salesman problem. We show how to modify the Christofides algorithm for Δ-TSP to obtain efficient approximation algorithms with constant approximation ratio for every instance of TSP that violates the triangle inequality by a multiplicative constant factor. This improves the result of Andreae and Bandelt [AB95].


Stability of approximation Traveling Salesman Problem 


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  1. [Ar97]
    S. Arora: Nearly linear time approximation schemes for Euclidean TSP and other geometric problems. In: Proc. 38th IEEE FOCS, 1997, pp. 554–563.Google Scholar
  2. [Ar98]
    S. Arora: Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. In: Journal of the ACM 45, No. 5 (Sep. 1998), pp. 753–782.zbMATHCrossRefMathSciNetGoogle Scholar
  3. [AB95]
    T. Andreae, H.-J. Bandelt: Performance guarantees for approximation algorithms depending on parametrized triangle inequalities. SIAM J. Discr. Math. 8 (1995), pp. 1–16.zbMATHCrossRefMathSciNetGoogle Scholar
  4. [BC93]
    D. P. Bovet, P. Crescenzi: Introduction to the Theory of Complexity, Prentice-Hall 1993.Google Scholar
  5. [BC99]
    M. A. Bender, C. Chekuri: Performance guarantees for the TSP with a parameterized triangle inequality. In: Proc. WADS’99, LNCS, to appear.Google Scholar
  6. [BHKSU00]
    H.-J. Böckenhauer, J. Hromkovič, R. Klasing, S. Seibert, W. Unger: An Improved Lower Bound on the Approximability of Metric TSP and Approximation Algorithms for the TSP with Sharpened Triangle Inequality (Extended Abstract). In: Proc. STACS’00, LNCS, to appear.Google Scholar
  7. [Chr76]
    N. Christofides: Worst-case analysis of a new heuristic for the traveling salesman problem. Technical Report 388, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, 1976.Google Scholar
  8. [Co71]
    S. A. Cook: The complexity of theorem proving procedures. In: Proc. 3rd ACM STOC, ACM 1971, pp. 151–158.Google Scholar
  9. [CK98]
    P. Crescenzi, V. Kann: A compendium of NP optimization problems.
  10. [CLR90]
    T. H. Cormen, C. E. Leiserson, R. L. Rivest: Introduction to algorithms. MIT Press, 1990.Google Scholar
  11. [En99]
    L. Engebretsen: An explicit lower bound for TSP with distances one and two. Extended abstract in: Proc. STACS’99, LNCS 1563, Springer 1999, pp. 373–382. Full version in: Electronic Colloquium on Computational Complexity, Report TR98-046 (1999).Google Scholar
  12. [GJ79]
    M. R. Garey, D. S. Johnson: Computers and vIntractability. A Guide to the Theory on NP-Completeness. W. H. Freeman and Company, 1979.Google Scholar
  13. [Håa97]
    J. Håstad: Some optimal inapproximability results. Extended abstract in: Proc. 29th ACM STOC, ACM 1997, pp. 1–10. Full version in: Electronic Colloquium on Computational Complexity, Report TR97-037, (1999).Google Scholar
  14. [Ho96]
    D. S. Hochbaum (Ed.): Approximation Algorithms for NP-hard Problems. PWS Publishing Company 1996.Google Scholar
  15. [Hr98]
    J. Hromkovič: Stability of approximation algorithms and the knapsack problem. Unpublished manuscript, RWTH Aachen, 1998.Google Scholar
  16. [IK75]
    O. H. Ibarra, C. E. Kim: Fast approximation algorithms for the knapsack and sum of subsets problem. J. of the ACM 22 (1975), pp. 463–468.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [Jo74]
    D. S. Johnson: Approximation algorithms for combinatorial problems. JCSS 9 (1974), pp. 256–278.zbMATHGoogle Scholar
  18. [Lo75]
    L. Lovász: On the ratio of the optimal integral and functional covers. Discrete Mathematics 13 (1975), pp. 383–390.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [LLRS85]
    E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, D. B. Shmoys (Eds.): The Traveling Salesman Problem. John Wiley & Sons, 1985.Google Scholar
  20. [Mi96]
    I. S. B. Mitchell: Guillotine subdivisions approximate polygonal subdivisions: Part II — a simple polynomial-time approximation scheme for geometric k-MST, TSP and related problems. Technical Report, Dept. of Applied Mathematics and Statistics, Stony Brook 1996.Google Scholar
  21. [MPS98]
    E.W. Mayr, H. J. Prömel, A. Steger (Eds.): Lectures on Proof Verification and Approximation Algorithms. LNCS 1967, Springer 1998.zbMATHGoogle Scholar
  22. [Pa77]
    Ch. Papadimitriou: The Euclidean traveling salesman problem is NP-complete. Theoretical Computer Science 4 (1977), pp. 237–244.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [Pa94]
    Ch. Papadimitriou: Computational Complexity, Addison-Wesley 1994.Google Scholar
  24. [PY93]
    Ch. Papadimitriou, M. Yannakakis: The traveling salesman problem with distances one and two. Mathematics of Operations Research 18 (1993), 1–11.zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Hans-Joachim Böckenhauer
    • 1
  • Juraj Hromkovič
    • 1
  • Ralf Klasing
    • 1
  • Sebastian Seibert
    • 1
  • Walter Unger
    • 1
  1. 1.Dept. of Computer Science I (Algorithms and Complexity)RWTH AachenAachenGermany

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