Fast sequential and randomised parallel algorithms for rigidity and approximate min k-cut

  • Sachin Patkar
  • H. Narayanan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 652)


In this paper we use new techniques based on flows and matroid theory to produce fast sequential and randomised parallel algorithms for two important classes of problems. The first class arises in the study of rigidity of graphs (also in the study of graph realizations). The second class of problems may be called Principal Partition related problems. We take a representative of this class, viz, the min k-cut problem and produce an RNC algorithm which solves this NP — hard problem within twice the optimal.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Crapo, H.: On the Generic Rigidity of Plane Frameworks, Research Report, No. 1278, INRIA, 1990.Google Scholar
  2. [2]
    Frank, A. and Tardos, E.: Generalized Polymatroids and Submodular Flows, Mathematical Programming, vol. 4, 1988, pp. 489–565.CrossRefGoogle Scholar
  3. [3]
    Gabow, H.N. and Westermann, H.H.: Forests, Frames and Games: Algorithms for Matroid sums and Applications, in Proc. 20 th STOC, 1988, pp. 407–421.Google Scholar
  4. [4]
    Hendrickson, B.: Conditions for Unique Graph Realizations, SIAM J. Computing, vol. 21, No. 1, 1992, pp. 65–84.CrossRefGoogle Scholar
  5. [5]
    Imai, H.: Network flow algorithms for lower truncated transversal polymatroids, J. of the Op. Research Society of Japan, vol. 26, 1983, pp. 186–210.Google Scholar
  6. [6]
    Laman, G.: On graphs and rigidity of plane skeletal structures, J. Engrg. Math., 4, 1970, pp. 331–340.CrossRefGoogle Scholar
  7. [7]
    Mulmuley, K., Vazirani, U.V. and Vazirani, V.V.: Matching is as easy as matrix inversion, Combinatorica, vol. 7, 1987, pp. 105–114.Google Scholar
  8. [8]
    Nakamura, M.: On the Representation of the Rigid Sub-systems of a Plane Link System, J. Op. Res. Soc. of Japan, vol. 29, No. 4, 1986, pp. 305–318.Google Scholar
  9. [9]
    Narayanan, H.: The Principal Lattice of Partitions of a Submodular function, Linear Algebra and its Applications, 144, 1991, pp. 179–216.CrossRefGoogle Scholar
  10. [10]
    Narayanan, H., Roy, S. and Patkar, S.: Min k-cut and the Principal Partition of a graph, in Proc. of the Second National Seminar on Theoretical Computer Science, India, 1992.Google Scholar
  11. [11]
    Narayanan, H., Saran, H. and Vazirani, V.V.: Fast parallel algorithms for Matroid Union, Arborescences and edge-disjoint spanning trees, in Proc. 3 rd ann. ACM-SIAM Symp. on Discrete Algorithms, 1992.Google Scholar
  12. [12]
    Patkar, S. and Narayanan, H.: Principal Lattice of Partitions of the Rank Function of a Graph, Technical Report VLSI-89-3, I.I.T. Bombay, 1989.Google Scholar
  13. [13]
    Patkar, S. and Narayanan, H.: Fast algorithm for the Principal Partition of a graph, in Proc. 11 th ann. symp. on Foundations of Software Technology and Theoretical Computer Science (FST & TCS-11), LNCS-560, 1991, pp. 288–306.Google Scholar
  14. [14]
    Patkar, S.:Investigations into the structure of graphs through the Principal Lattice of Partitions approach, Ph.D. thesis, Dept. of Computer Sci. and Engg., IIT Bombay, INDIA, 1992.Google Scholar
  15. [15]
    Saran, H. and Vazirani, V.V.: Finding a k-Cut within Twice the Optimal, Proc. 32 nd annual Symp. on the Foundations of Computer Science, 1991.Google Scholar
  16. [16]
    Welsh, D. J. A.: Matroid Theory, Academic Press, New York, 1976.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • Sachin Patkar
    • 1
  • H. Narayanan
    • 2
  1. 1.Dept. of Computer Science and Engg.IIT BombayBombayIndia
  2. 2.Dept. of Electrical Engg.IIT BombayBombayIndia

Personalised recommendations