Fast On-Line/Off-Line Algorithms for Optimal Reinforcement of a Network and Its Connections with Principal Partition

  • Sachin B. Patkar
  • H. Narayanan
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1974)


The problem of computing the strength and performing optimal reinforcement for an edge-weighted graph G(V, E,w) is well-studied [1],[2],[3],[6],[7],[9]. In this paper, we present fast (sequential linear time and parallel logarithmic time) on-line algorithms for optimally reinforcing the graph when the reinforcement material is available continuosly online. These are first on-line algortithms for this problem. Although we invest some time in preprocessing the graph before the start of our algorithms, it is also shown that the output of our on-line algorithms is as good as that of the off-line algorithms, making our algorithms viable alternatives to the fastest off-line algorithms in situtations when a sequence of more than O(|V|) reinforcement problems need to be solved. In such a situation the time taken for preprocessing the graph is less that the time taken for all the invocations of the fastest off-line algorithms. Thus our algorithms are also efficient in the general sense. The key idea is to make use of the theory of Principal Partition of a Graph. Our results can be easily generalized to the general setting of principal partition of nondecreasing submodular functions.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Barahona, F.: Separating from the dominant of spanning tree polytope,Oper. Res. Letters, vol. 12, 1992, pp.201–203.zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cheng, E. and Cunningham, W.: A faster algorithm for computing thestrength of a network, Information Processing Letters, vol. 49, 1994, pp.209–212.zbMATHCrossRefGoogle Scholar
  3. 3.
    Cunningham, W.: Optimal attack and reinforcement of a network, JACM,vol. 32, no. 3, 1985, pp.549–561.zbMATHMathSciNetGoogle Scholar
  4. 4.
    Dinkelbach, W.: On nonlinear fractional programming, Management Sci.,vol. 13, 1967, pp.492–498.MathSciNetCrossRefGoogle Scholar
  5. 5.
    Edmonds, J.: Submodular functions, matroids and certain polyhedra, Proc.Calgary Intl. conference on Combinatorial Structures, 1970, pp.69–87.Google Scholar
  6. 6.
    Fujishige, S.:Submodular Functions and Optimization, Annals of DiscreteMathematics, North Holland, 1991.Google Scholar
  7. 7.
    Gabow, H.N.: Algorithms for graphic polymatroids and parametric s-sets J.Algorithms, vol. 26, 1998, pp.48–86.zbMATHMathSciNetGoogle Scholar
  8. 8.
    Gallo, G., Grigoriadis, M. and Tarjan, R.E.: A fast parametric network flow algorithm, SIAM J. of Computing, vol. 18, 1989, pp.30–55.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gusfield, D.: Computing the strength of a graph, SIAM J. of Computing,vol. 20, 1991, pp.639–654.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Imai, H.: Network flow algorithms for lower truncated transversal polymatroids,J. of the Op. Research Society of Japan, vol. 26, 1983, pp. 186–210.zbMATHGoogle Scholar
  11. 11.
    Iri, M. and Fujishige, S.: Use of matroid theory in operations research, circuitsand systems theory, Int. J. Systems Sci.,vol. 12, no. 1, 1981, pp. 27–54.zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Itai, A. and Rodeh, M.: The multi-tree approach to reliability in distributednetworks, in Proc. 25th ann. symp. FOCS, 1984, pp. 137–147.Google Scholar
  13. 13.
    Narayanan, H.:Theory of matroids and network analysis, Ph.D. thesis, Departmentof Electrical Engineering, I.I.T. Bombay, 1974.Google Scholar
  14. 14.
    Narayanan, H.: The principal lattice of partitions of a submodular function,Linear Algebra and its Applications, 144, 1991, pp. 179–216.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Narayanan, H., Roy, S. and Patkar, S.B.: Approximation Algorithms formin-k-overlap Problems, using the Principal Lattice of Partitions Approach,J. of Algorithms, vol. 21, 1996, pp. 306–330.zbMATHMathSciNetGoogle Scholar
  16. 16.
    Narayanan, H.: Submodular Functions and Electrical Networks, Annals ofDiscrete Mathematics-54, North Holland, 1997.Google Scholar
  17. 17.
    Patkar, S. and Narayanan, H.: Principal lattice of partitions of submodularfunctions on graphs: fast algorithms for principal partition and genericrigidity, in Proc. of the 3rd ann. Int. Symp. on Algorithms and Computation,(ISAAC), LNCS-650, Japan, 1992, pp. 41–50.Google Scholar
  18. 18.
    Patkar, S. and Narayanan, H.: Fast On-line/Off-line Algorithms for OptimalReinforcement of a Network and its Connections with Principal Partition,Technical Report, Industrial Mathematics Group, Department of Mathematics,IIT Bombay, available from authors via e-mail, 2000.Google Scholar
  19. 19.
    Tomizawa, N.: Strongly irreducible matroids and principal partition of amatroid into strongly irreducible minors (in Japanese), Transactions of theInstitute of Electronics and Communication Engineers of Japan, vol. J59A,1976, pp. 83–91.MathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Sachin B. Patkar
    • 1
  • H. Narayanan
    • 2
  1. 1.Department of MathematicsIndian Institute of Technology - BombayMumbaiIndia
  2. 2.Department of Electrical Engg.Indian Institute of Technology - BombayMumbaiIndia

Personalised recommendations