Greedy Edge-Disjoint Paths in Complete Graphs

  • Paz Carmi
  • Thomas Erlebach
  • Yoshio Okamoto
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2880)


The maximum edge-disjoint paths problem (MEDP) is one of the most classical NP-hard problems. We study the approximation ratio of a simple and practical approximation algorithm, the shortest-path-first greedy algorithm (SGA), for MEDP in complete graphs. Previously, it was known that this ratio is at most 54. Adapting results by Kolman and Scheideler [Proceedings of SODA, 2002, pp. 184–193], we show that SGA achieves approximation ratio 8F+1 for MEDP in undirected graphs with flow number F and, therefore, has approximation ratio at most 9 in complete graphs. Furthermore, we construct a family of instances that shows that SGA cannot be better than a 3-approximation algorithm. Our upper and lower bounds hold also for the bounded-length greedy algorithm, a simple on-line algorithm for MEDP.


Approximation algorithm Greedy algorithm Shortening lemma 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Carmi, P., Erlebach, T., Okamoto, Y.: Greedy edge-disjoint paths in complete graphs. TIK-Report 155, Computer Engineering and Networks Laboratory (TIK), ETH Zürich (February 2003)Google Scholar
  2. 2.
    Chekuri, C., Khanna, S.: Edge disjoint paths revisited. In: Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2003), pp. 628–637 (2003)Google Scholar
  3. 3.
    Erlebach, T.: Approximation algorithms and complexity results for path problems in trees of rings. TIK-Report 109, Computer Engineering and Networks Laboratory (TIK), ETH Zürich (June 2001)Google Scholar
  4. 4.
    Erlebach, T., Vukadinović, D.: New results for path problems in generalized stars, complete graphs, and brick wall graphs. In: Freivalds, R. (ed.) FCT 2001. LNCS, vol. 2138, pp. 483–494. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  5. 5.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York (1979)zbMATHGoogle Scholar
  6. 6.
    Garg, N., Vazirani, V.V., Yannakakis, M.: Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18, 3–20 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Guruswami, V., Khanna, S., Rajaraman, R., Shepherd, B., Yannakakis, M.: Nearoptimal hardness results and approximation algorithms for edge-disjoint paths and related problems. In: Proceedings of the 31st Annual ACM Symposium on Theory of Computing (STOC 1999), pp. 19–28 (1999)Google Scholar
  8. 8.
    Kleinberg, J.M.: Approximation algorithms for disjoint paths problems. Ph.D. thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology (1996)Google Scholar
  9. 9.
    Kolliopoulos, S.G., Stein, C.: Approximating disjoint-path problems using greedy algorithms and packing integer programs. In: Bixby, R.E., Boyd, E.A., Ríos-Mercado, R.Z. (eds.) IPCO 1998. LNCS, vol. 1412, pp. 153–168. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  10. 10.
    Kolman, P., Scheideler, C.: Improved bounds for the unsplittable flow problem. In: Proceedings of the 13th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2002), pp. 184–193 (2002)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2003

Authors and Affiliations

  • Paz Carmi
    • 1
  • Thomas Erlebach
    • 2
  • Yoshio Okamoto
    • 3
  1. 1.Department of Computer ScienceBen-Gurion University of the NegevIsrael
  2. 2.Computer Engineering and Networks Laboratory, Department of Information Technology and Electrical EngineeringETH ZürichSwitzerland
  3. 3.Institute of Theoretical Computer Science, Department of Computer ScienceETH ZürichSwitzerland

Personalised recommendations