, Volume 57, Issue 3, pp 517–537

A Preemptive Algorithm for Maximizing Disjoint Paths on Trees



We consider the on-line version of the maximum vertex disjoint path problem when the underlying network is a tree. In this problem, a sequence of requests arrives in an on-line fashion, where every request is a path in the tree. The on-line algorithm may accept a request only if it does not share a vertex with a previously accepted request. The goal is to maximize the number of accepted requests. It is known that no on-line algorithm can have a competitive ratio better than Ω(log n) for this problem, even if the algorithm is randomized and the tree is simply a line. Obviously, it is desirable to beat the logarithmic lower bound. Adler and Azar (Proc. of the 10th ACM-SIAM Symposium on Discrete Algorithm, pp. 1–10, 1999) showed that if preemption is allowed (namely, previously accepted requests may be discarded, but once a request is discarded it can no longer be accepted), then there is a randomized on-line algorithm that achieves constant competitive ratio on the line. In the current work we present a randomized on-line algorithm with preemption that has constant competitive ratio on any tree. Our results carry over to the related problem of maximizing the number of accepted paths subject to a capacity constraint on vertices (in the disjoint path problem this capacity is 1). Moreover, if the available capacity is at least 4, randomization is not needed and our on-line algorithm becomes deterministic.


Online algorithms Disjoint paths Disjoint paths on trees Admission control Call control 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Adler, R., Azar, Y.: Beating the logarithmic lower bound: randomized preemptive disjoint paths and call control algorithms. In: Proc. of the 10th ACM-SIAM Symposium on Discrete Algorithms, pp. 1–10 (1999) Google Scholar
  2. 2.
    Alon, N., Arad, U., Azar, Y.: Independent sets in hypergraphs with applications to routing via fixed paths. In: Proc. 2nd Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), pp. 16–27 (1999) Google Scholar
  3. 3.
    Andrews, M., Chuzhoy, J., Khanna, S., Zhang, L.: Hardness of the undirected edge-disjoint paths problem with congestion. In: Proceedings 46th Annual IEEE Symposium on Foundations of Computer Science, pp. 226–244 (2005) Google Scholar
  4. 4.
    Aspnes, J., Azar, Y., Fiat, A., Plotkin, S., Waarts, O.: On-line routing of virtual circuits with applications to load balancing and machine scheduling. J. ACM 44(3), 486–504 (1997). Also in Proc. 25th ACM STOC, 1993, pp. 623–631 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Awerbuch, B., Azar, Y., Plotkin, S.: Throughput-competitive online routing. In: 34th IEEE Symposium on Foundations of Computer Science, pp. 32–40 (1993) Google Scholar
  6. 6.
    Awerbuch, B., Bartal, Y., Fiat, A., Rosén, A.: Competitive non-preemptive call control. In: Proc. of 5th ACM-SIAM Symposium on Discrete Algorithms, pp. 312–320 (1994) Google Scholar
  7. 7.
    Awerbuch, B., Gawlick, R., Leighton, T., Rabani, Y.: On-line admission control and circuit routing for high performance computation and communication. In: Proc. 35th IEEE Symp. on Found. of Comp. Science, pp. 412–423 (1994) Google Scholar
  8. 8.
    Awerbuch, B., Azar, Y., Fiat, A., Leonardi, S., Rosen, A.: On-line competitive algorithms for call admission in optical networks. In: Proc. 4th Annual European Symposium on Algorithms, pp. 431–444 (1996) Google Scholar
  9. 9.
    Bartal, Y., Fiat, A., Leonardi, S.: Lower bounds for on-line graph problems with application to on-line circuit and optical routing. In: Proc. 28th ACM Symp. on Theory of Computing, pp. 531–540 (1996) Google Scholar
  10. 10.
    Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, Cambridge (1998) MATHGoogle Scholar
  11. 11.
    Cormen, T.T., Leiserson, C.E., Rivest, R.L.: Introduction to Algorithms. MIT Press, Cambridge (1990) MATHGoogle Scholar
  12. 12.
    Garay, J., Gopal, I., Kutten, S., Mansour, Y., Yung, M.: Efficient on-line call control algorithms. J. Algorithms 23, 180–194 (1997). Also in Proc. 2nd Annual Israel Conference on Theory of Computing and Systems (1993) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. In: ALGORITHMICA, vol. 18, pp. 3–20 (1997) Google Scholar
  14. 14.
    Kleinberg, J., Tardos, E.: Disjoint paths in densely embedded graphs. In: Proc. 36th IEEE Symp. on Found. of Comp. Science, pp. 52–61 (1995) Google Scholar
  15. 15.
    Leonardi, S.: On-line network routing. In: Fiat, A., Woeginger, G. (eds.) Online Algorithms—The State of the Art, pp. 242–267. Springer, Berlin (1998). Chap. 11 Google Scholar
  16. 16.
    Leonardi, S., Marchetti-Spaccamela, A., Presciutti, A., Rosén, A.: On-line randomized call control revisited. In: Proc. 9th ACM-SIAM Symp. on Discrete Algorithms, pp. 323–332 (1998) Google Scholar
  17. 17.
    Lipton, R.J., Tomkins, A.: Online interval scheduling. In: Proc. of the 5th ACM-SIAM Symposium on Discrete Algorithms, pp. 302–311 (1994) Google Scholar
  18. 18.
    Raghavan, P., Thompson, C.D.: Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7(4), 365–374 (1987) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Tel Aviv UniversityTel AvivIsrael
  2. 2.Microsoft ResearchRedmondUSA
  3. 3.Weizmann InstituteRehovotIsrael

Personalised recommendations