Advertisement

Algorithmica

, Volume 18, Issue 1, pp 3–20 | Cite as

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

  • N. Garg
  • V. V. Vazirani
  • M. Yannakakis
Article

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.

Key Words

Integral multicommodity flow Multicut Approximation algorithm MAX SNP-hard 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [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.Google Scholar
  2. [2]
    R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem.J. Algorithms, 2:198–203, 1981.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [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.Google Scholar
  4. [4]
    C. Berge.Graph and Hypergraphs. North-Holland, Amsterdam, 1976.Google Scholar
  5. [5]
    R.E. Bixby and D.K. Wagner. An almost linear time algorithm for graph realization.Math. Oper. Res., 13:99–123, 1988.zbMATHMathSciNetCrossRefGoogle Scholar
  6. [6]
    B.V. Cherkasskij. Solution of aproblem of multicommodity flows in a network (in Russian).Mat. Metody, 13:143–151, 1977.zbMATHGoogle Scholar
  7. [7]
    S. Chopra and M.R. Rao. On the multiway cut polyhedron.Networks, 21:51–89, 1991.zbMATHMathSciNetGoogle Scholar
  8. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  9. [9]
    S. Even, A. Itai, and A. Shamir. On the complexity of timetable and multicommodity flow problems.SIAM J. Comput., 5:691–703, 1976.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [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. [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.Google Scholar
  12. [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.Google Scholar
  13. [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.Google Scholar
  14. [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.zbMATHMathSciNetCrossRefGoogle Scholar
  15. [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. [16]
    M. Grotschel, L. Lovasz, and A. Schrijver.Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin, 1988.Google Scholar
  17. [17]
    T.C. Hu,Integer Programming and Network Flows. Addison-Wesley, Reading, MA, 1969.zbMATHGoogle Scholar
  18. [18]
    V. Kann. On the approximability of NP-complete optimization problems. Ph.D. Thesis, Royal Institute of Technology, Stockholm, 1992.Google Scholar
  19. [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.Google Scholar
  20. [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. [21]
    L. Lovász. On some connectivity properties of eulerian graphs.Acta Math. Akad. Sci. Hungar., 28:129–138, 1976.zbMATHCrossRefGoogle Scholar
  22. [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.Google Scholar
  23. [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.zbMATHMathSciNetGoogle Scholar
  24. [24]
    W. Mader. Uber die maximalzahl kantendisjunkter a-wege.Arch. Math., 30:325–336, 1978.zbMATHMathSciNetCrossRefGoogle Scholar
  25. [25]
    C.H. Papadimitriou and M. Yannakakis. Optimization, approximation and complexity classes.J. Comput. System Sci., 43:425–440, 1991.zbMATHMathSciNetCrossRefGoogle Scholar
  26. [26]
    N. Robertson and P.D. Seymour. Graph minors XIII: The disjoint path problem.J. Combin. Theory Ser. B, 63:65–110, 1995.zbMATHMathSciNetCrossRefGoogle Scholar
  27. [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. [28]
    A. Srivastav and P. Stangier. Integer multicommodity flows with reduced demands.Proceedings European Symposium on Algorithms, pages 360–372, 1993.Google Scholar
  29. [29]
    W.T. Tutte. An algorithm for determining whether a given binary matroid is graphic.Proc. Amer. Math. Soc., 11:905–917, 1960.MathSciNetCrossRefGoogle Scholar
  30. [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.Google Scholar
  31. [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.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1997

Authors and Affiliations

  • N. Garg
    • 1
  • V. V. Vazirani
    • 1
  • M. Yannakakis
    • 2
  1. 1.Department of Computer Science and EngineeringIndian Institute of TechnologyNew DelhiIndia
  2. 2.AT&T Bell LaboratoriesMurray HillUSA

Personalised recommendations