Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content
Springer Nature Link
Log in

Primal-dual approximation algorithms for integral flow and multicut in trees

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

We study the maximum integral multicommodity flow problem and the minimum multicut problem restricted to trees. This restriction is quite rich and contains as special cases classical optimization problems such as matching and vertex cover for general graphs. It is shown that both the maximum integral multicommodity flow and the minimum multicut problem are NP-hard and MAX SNP-hard on trees, although the maximum integral flow can be computed in polynomial time if the edges have unit capacity. We present an efficient algorithm that computes a multicut and integral flow such that the weight of the multicut is at most twice the value of the flow. This gives a 2-approximation algorithm for minimum multicut and a 1/2-approximation algorithm for maximum integral multicommodity flow in trees.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and the hardness of approximation problems.Proceedings, 33rd IEEE Symposium on Foundations of Computer Science, pages 14–23, 1992.

  2. R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem.J. Algorithms, 2:198–203, 1981.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Bellare, S. Goldwasser, C. Lund, and A. Russell. Efficient probabilistically checkable proofs and applications to approximation.Proceedings, 25th Annual ACM Symposium on Theory of Computing, pages 294–305, 1993.

  4. C. Berge.Graph and Hypergraphs. North-Holland, Amsterdam, 1976.

    Google Scholar 

  5. R.E. Bixby and D.K. Wagner. An almost linear time algorithm for graph realization.Math. Oper. Res., 13:99–123, 1988.

    Article  MATH  MathSciNet  Google Scholar 

  6. B.V. Cherkasskij. Solution of aproblem of multicommodity flows in a network (in Russian).Mat. Metody, 13:143–151, 1977.

    MATH  Google Scholar 

  7. S. Chopra and M.R. Rao. On the multiway cut polyhedron.Networks, 21:51–89, 1991.

    MATH  MathSciNet  Google Scholar 

  8. E. Dahlhaus, D.S. Johnson, C.H. Papadimitriou, P.D. Seymour, and M. Yannakakis. The complexity of multiterminal cuts.SIAM J. Comput., 23:864–894, 1994. Preliminary version appeared under the title, The complexity of multiway cuts,Proceedings, 24th Annual ACM Symposium on Theory of Computing, pages 241–251, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  9. S. Even, A. Itai, and A. Shamir. On the complexity of timetable and multicommodity flow problems.SIAM J. Comput., 5:691–703, 1976.

    Article  MATH  MathSciNet  Google Scholar 

  10. A. Frank. Packing paths, circuits and cuts—a survey. In B. Korte, L. Lovasz, H.J. Promel, and A. Schrijver, editors,Paths, Flows and VLSI-Layout, pages 47–100. Algorithms and Combinatorics, volume 9. Springer-Verlag, Berlin, 1991.

    Google Scholar 

  11. H.N. Gabow. An efficient reduction technique for degree-constrained subgraph and bidirected network flow problems.Proceedings, 15th Annual ACM Symposium on Theory of Computing, pages 448–456, 1983.

  12. N. Garg, V.V. Vazirani, and M. Yannakakis. Approximate max-flow min-(multi)cut theorems and their applications.Proceedings, 25th Annual ACM Symposium on Theory of Computing, pages 698–707, 1993.

  13. N. Garg, V.V. Vazirani, and M. Yannakakis. Approximation algorithms for multiway cuts in node-weighted and directed graphs.Proceedings, 21st International Colloquium on Automata, Languages and Programming, pages 487–498, 1994.

  14. M.X. Goemans and D.P. Williamson. A general approximation technique for constrained forest problems.SIAM J. Comput., 24:296–317, 1995. Preliminary version inProceedings, 3rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 307–316, 1992.

    Article  MATH  MathSciNet  Google Scholar 

  15. M.X. Goemans and D.P. Williamson. The primal-dual method for approximation algorithms and its application to network design problems. In D. Hochbaum, editor,Approximation Algorithms for NP-hard Problems, pages 144–191. PWS Publishing, Boston, 1995.

    Google Scholar 

  16. M. Grotschel, L. Lovasz, and A. Schrijver.Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin, 1988.

    Google Scholar 

  17. T.C. Hu,Integer Programming and Network Flows. Addison-Wesley, Reading, MA, 1969.

    MATH  Google Scholar 

  18. V. Kann. On the approximability of NP-complete optimization problems. Ph.D. Thesis, Royal Institute of Technology, Stockholm, 1992.

    Google Scholar 

  19. P. Klein, A. Agrawal, R. Ravi, and S. Rao. Approximation through multicommodity flow.Proceedings 31st IEEE Symposium on Foundations of Computer Science pages 726–737, 1990.

  20. E. Korach and M. Penn. Tight integral duality gap in the Chinese postman problem. Technical Report, Computer Science Department, Israel Institute of Technology, Haifa, 1989.

    Google Scholar 

  21. L. Lovász. On some connectivity properties of eulerian graphs.Acta Math. Akad. Sci. Hungar., 28:129–138, 1976.

    Article  MATH  Google Scholar 

  22. F.T. Leighton and S. Rao. An approximate max-flow min-cut theorem for uniform multicommodity flow problems with application to approximation algorithms.Proceedings 29th Symposium on Foundations of Computer Science, pages 422–431, 1988.

  23. C. Lund and M. Yannakakis. On the hardness of approximating minimization problems.J. Assoc. Comput. Mach., 41(5):960–981, 1994. Preliminary version appeared inProceedings 25th Annual ACM Symposium on Theory of Computing, pages 286–293, 1993.

    MATH  MathSciNet  Google Scholar 

  24. W. Mader. Uber die maximalzahl kantendisjunkter a-wege.Arch. Math., 30:325–336, 1978.

    Article  MATH  MathSciNet  Google Scholar 

  25. C.H. Papadimitriou and M. Yannakakis. Optimization, approximation and complexity classes.J. Comput. System Sci., 43:425–440, 1991.

    Article  MATH  MathSciNet  Google Scholar 

  26. N. Robertson and P.D. Seymour. Graph minors XIII: The disjoint path problem.J. Combin. Theory Ser. B, 63:65–110, 1995.

    Article  MATH  MathSciNet  Google Scholar 

  27. A. Schrijver. Homotopic routing methods. In B. Korte, L. Lovász, H.J. Promel, and A. Schrijver, editors,Paths, Flows and VLSI-Layout, pages 329–371. Algorithms and Combinatorics, volume 9. Springer-Verlag, Berlin, 1991.

    Google Scholar 

  28. A. Srivastav and P. Stangier. Integer multicommodity flows with reduced demands.Proceedings European Symposium on Algorithms, pages 360–372, 1993.

  29. W.T. Tutte. An algorithm for determining whether a given binary matroid is graphic.Proc. Amer. Math. Soc., 11:905–917, 1960.

    Article  MathSciNet  Google Scholar 

  30. D.P. Williamson, M.X. Goemans, M. Mihail, and V.V. Vazirani. A primal-dual approximation algorithm for generalized steiner network problems.Proceedings, 25th Annual ACM Symposium on Theory of Computing, pages 708–717, 1993.

  31. M. Yannakakis, P.C. Kanellakis, S.C. Cosmadakis, and C. H. Papadimitriou. Cutting and partitioning a graph after a fixed pattern. InAutomata, Languages and Programming, pages 712–722. Lecture notes in Computer Science, volume 154. Springer-Verlag, Berlin, 1983.

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science and Engineering, Indian Institute of Technology, New Delhi, India

    N. Garg & V. V. Vazirani

  2. AT&T Bell Laboratories, 07974, Murray Hill, NJ, USA

    M. Yannakakis

Authors
  1. N. Garg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. V. Vazirani
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. Yannakakis
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by M. X. Goemans.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Garg, N., Vazirani, V.V. & Yannakakis, M. Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18, 3–20 (1997). https://doi.org/10.1007/BF02523685

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02523685

Key Words

Search

Navigation