A Constant Approximation Algorithm for the a priori Traveling Salesman Problem

  • David Shmoys
  • Kunal Talwar
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5035)


One of the interesting recent developments in the design and analysis of approximation algorithms has been in the area of algorithms for discrete stochastic optimization problems. In this domain, one is given not just one input, but rather a probability distribution over inputs, and yet the aim is to design an algorithm that has provably good worst-case performance, that is, for any probability distribution over inputs, the objective function value of the solution found by the algorithm must be within a specified factor of the optimal value.

The a priori traveling salesman problem is a prime example of such a stochastic optimization problem. One starts with the standard traveling salesman problem (in which one wishes to find the shortest tour through a given set of points N), and then considers the possibility that only a subset A of the entire set of points is active. The active set is given probabilistically; that is, there is a probability distribution over the subsets of N, which is given as part of the input. The aim is still to compute a tour through all points in N, but in order to evaluate its cost, we instead compute the expectation of the length of this tour after shortcutting it to include only those points in the active set A (where the expectation is computed with respect to the given probability distribution). The goal is to compute a “master tour” for which this expectation is minimized. This problem was introduced in the doctoral theses of Jaillet and Bertsimas, who gave asymptotic analyses when the distances between points in the input set are also given probabilistically.

In this paper, we restrict attention to the so-called “independent activation” model in which we assume that each point j is active with a given probability pj, and that these independent random events. For this setting, we give a 8-approximation algorithm, a polynomial-time algorithm that computes a tour whose a priori TSP objective function value is guaranteed to be within a factor of 8 of optimal (and a randomized 4-approximation algorithm, which produces a tour of expected cost within a factor of 4 of optimal). This is the first constant approximation algorithm for this model.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Jaillet, P.: Probabilistic traveling salesman problems. Technical Report 185, Operations Research Center, MIT (1985)Google Scholar
  2. 2.
    Jaillet, P.: A priori solution of a traveling salesman problem in which a random subset of the customers are visited. Operations Research 36, 929–936 (1988)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bertsimas, D.: Probabilistic Combinatorial Optimization Problems. PhD thesis, MIT, Cambridge, Mass (1988)Google Scholar
  4. 4.
    Bertsimas, D.J., Jaillet, P., Odoni, A.R.: A priori optimization. Operations Research 38(6), 1019–1033 (1990)MATHMathSciNetGoogle Scholar
  5. 5.
    Schalekamp, F., Shmoys, D.B.: Algorithms for the universal and a priori TSP. Operations Research Letters (in press, 2007)Google Scholar
  6. 6.
    Bartholdi III, J.J., Platzman, L.K.: An O(N log N) planar travelling salesman heuristic based on spacefilling curves. Operations Research Letters 1(4), 121–125 (1981/82)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Hajiaghayi, M.T., Kleinberg, R., Leighton, T.: Improved lower and upper bounds for universal tsp in planar metrics. In: Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 649–658. ACM Press, New York (2006)CrossRefGoogle Scholar
  8. 8.
    Gupta, A., Kumar, A., Pál, M., Roughgarden, T.: Approximation via cost-sharing: a simple approximation algorithm for the multicommodity rent-or-buy problem. In: Proceedings of the 44th Anuual IEEE Symposium on Foundations of Computer Science, Silver Spring, MD, pp. 606–615. IEEE Computer Society, Los Alamitos (2003)CrossRefGoogle Scholar
  9. 9.
    Gupta, A., Pál, M., Ravi, R., Sinha, A.: Boosted sampling: approximation algorithms for stochastic optimization. In: Proceedings of the 36th Annual ACM Symposium on Theory of Computing, pp. 265–274 (2004)Google Scholar
  10. 10.
    Williamson, D.P., van Zuylen, A.: A simpler and better derandomization of an approximation algorithm for single source rent-or-buy. Operations Research Letters 35, 707–712 (2007)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Garg, N., Gupta, A., Leonardi, S., Sankowski, P.: Private communication (2007)Google Scholar
  12. 12.
    Garg, N., Gupta, A., Leonardi, S., Sankowski, P.: Stochastic analyses for online combinatorial optimization problems. In: Proceedings of the 19th Anuual ACM-SIAM Symposium on Discrete Algorithms, pp. 942–951. ACM and SIAM, New York and Philadelphia (2008)Google Scholar
  13. 13.
    Lawler, E.L., Lenstra, J.K., Rinnooy Kan, A.H.G., Shmoys, D.B. (eds.): The Traveling Salesman Problem. Wiley, Chichester (1985)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • David Shmoys
    • 1
  • Kunal Talwar
    • 2
  1. 1.Cornell UniversityIthaca
  2. 2.Microsoft ResearchMountain View

Personalised recommendations