A Probabilistic Analysis of Christofides’ Algorithm

  • Markus Bläser
  • Konstatinos Panagiotou
  • B. V. Raghavendra Rao
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)


Christofides’ algorithm is a well known approximation algorithm for the metric travelling salesman problem. As a first step towards obtaining an average case analysis of Christofides’ algorithm, we provide a probabilistic analysis for the stochastic version of the algorithm for the Euclidean traveling salesman problem, where the input consists of n randomly chosen points in [0,1] d . Our main result provides bounds for the length of the computed tour that hold almost surely. We also provide an experimental evaluation of Christofides’s algorithm.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arora, S.: Polynomial time approximation schemes for euclidean traveling salesman and other geometric problems. J. ACM 45(5), 753–782 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Avis, D., Davis, B., Steele, J.M.: Probabilistic analysis of a greedy heuristic for euclidean matching. Probability in the Engineering and Informational Sciences 2(02), 143–156 (1988)zbMATHCrossRefGoogle Scholar
  3. 3.
    Baltz, A., Dubhashi, D.P., Srivastav, A., Tansini, L., Werth, S.: Probabilistic analysis for a multiple depot vehicle routing problem. Random Struct. Algorithms 30(1-2), 206–225 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Beardwood, J., Halton, J.H., Hammersley, J.M.: The shortest path through many points. Proc. Cambridge Philos. Soc. 55, 299–327 (1959)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bertsimas, D.J., van Ryzin, G.: An asymptotic determination of the minimum spanning tree and minimum matching constants in geometrical probability. Operations Research Letters 9(4), 223–231 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Christofides, N.: Worst-case analysis of a new heuristic for the traveling salesman problem. Report 388, Graduate School of Industrial Administration, Carnegie Mellon University (1976)Google Scholar
  7. 7.
    Deineko, V., Tiskin, A.: Fast minimum-weight double-tree shortcutting for metric tsp: Is the best one good enough? J. Exp. Algorithmics 14, 4.6–4.16 (2010)Google Scholar
  8. 8.
    Frieze, A.M., Yukich, J.E.: Probabilistic analysis of the traveling salesman problem. In: Gutin, G., Punnen, A. (eds.) The Traveling Salesman Problem and Its Variations, pp. 257–308. Kluwer Academic Publisher (2002)Google Scholar
  9. 9.
    Goemans, M.X., Bertsimas, D.J.: Probabilistic analysis of the held and karp lower bound for the euclidean traveling salesman problem. Mathematics of Operations Research 16(1), 72–89 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Gutin, G., Punnen, A.P. (eds.): The traveling salesman problem and its variations. Combinatorial Optimization, vol. 12. Kluwer Academic Publishers (2002)Google Scholar
  11. 11.
    Johnson, D.S., McGeoch, L.A., Rothberg, E.E.: Asymptotic experimental analysis for the held-karp traveling salesman bound. In: SODA 1996, pp. 341–350 (1996)Google Scholar
  12. 12.
    Karp, R.M.: Probabilistic analysis of partitioning algorithms for the traveling-salesman problem in the plane. Math. of Operat. Research 2(3), 209–224 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Mitchell, J.: Guillotine subdivisions approximate polygonal subdivisions: A simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM J. Comput. 28(4), 1298–1309 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Rhee, W.T.: On the travelling salesperson problem in many dimensions. Random Struct. Algorithms 3(3), 227–234 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Rhee, W.T.: A matching problem and subadditive euclidean functionals. Ann. Appl. Probab. 3(3), 794–801 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Steele, J.M.: Subadditive Euclidean functionals and nonlinear growth in geometric probability. Ann. Probab. 9(3), 365–376 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  17. 17.
    Steele, J.M.: Growth rates of Euclidean minimal spanning trees with power weighted edges. Ann. Probab. 16, 1767–1787 (1988)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Steele, J.M.: Probability Theory and Combinatorial Optimization. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 69. SIAM (1997)Google Scholar
  19. 19.
    Yukich, J.E.: Probability Theory of Classical Euclidean Optimization Problems. Lecture Notes in Mathematics, vol. 1675. Springer (1998)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Markus Bläser
    • 1
  • Konstatinos Panagiotou
    • 2
  • B. V. Raghavendra Rao
    • 1
  1. 1.Department of Computer ScienceSaarland UniversityGermany
  2. 2.Department of MathematicsUniversity of MunichGermany

Personalised recommendations