Greedy Approximation Algorithms for Finding Dense Components in a Graph

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1913)


We study the problem of finding highly connected subgraphs of undirected and directed graphs. For undirected graphs, the notion of density of a subgraph we use is the average degree of the subgraph. For directed graphs, a corresponding notion of density was introduced recently by Kannan and Vinay. This is designed to quantify highly connectedness of substructures in a sparse directed graph such as the web graph. We study the optimization problems of finding subgraphs maximizing these notions of density for undirected and directed graphs. This paper gives simple greedy approximation algorithms for these optimization problems. We also answer an open question about the complexity of the optimization problem for directed graphs.


Directed Graph Greedy Algorithm Undirected Graph Dual Solution Dense Subgraph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Y. Asahiro and K. Iwama. Finding Dense Subgraphs. Proc. 6th International Symposium on Algorithms and Computation (ISAAC), LNCS 1004, 102–111 (1995).Google Scholar
  2. 2.
    Y. Asahiro, K. Iwama, H. Tamaki and T. Tokuyama. Greedily Finding a Dense Subgraph. Journal of Algorithms, 34(2):203–221 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    P. Drineas, A. Frieze, R. Kannan, S. Vempala and V. Vinay. Clustering in Large Graphs and Matrices. Proc. 10th Annual ACM-SIAM Symposium on Discrete Algorithms, 291–299 (1999).Google Scholar
  4. 4.
    U. Feige, G. Kortsarz and D. Peleg. The Dense k-Subgraph Problem. Algorithmica, to appear. Preliminary version in Proc. 34th Annual IEEE Symposium on Foundations of Computer Science, 692–701 (1993).Google Scholar
  5. 5.
    U. Feige and M. Seltser. On the Densest k-Subgraph Problem. Weizmann Institute Technical Report CS 97-16 (1997).Google Scholar
  6. 6.
    A. Frieze, R. Kannan and S. Vempala. Fast Monte-Carlo Algorithms for Finding Low Rank Approximations. Proc. 39th Annual IEEE Symposium on Foundations of Computer Science, 370–378 (1998).Google Scholar
  7. 7.
    G. Gallo, M. D. Grigoriadis, and R. Tarjan. A Fast Parametric Maximum Flow Algorithm and Applications. SIAM J. on Comput., 18:30–55 (1989).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    D. Gibson, J. Kleinberg and P. Raghavan. Inferring web communities from Web topology. Proc. HYPERTEXT, 225–234 (1998).Google Scholar
  9. 9.
    R. Kannan and V. Vinay. Analyzing the Structure of Large Graphs. manuscript, August 1999.Google Scholar
  10. 10.
    J. Kleinberg. Authoritative sources in hypertext linked environments. Proc. 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 668–677 (1998).Google Scholar
  11. 11.
    J. Kleinberg, R. Kumar, P. Raghavan, S. Rajagopalan and A. Tomkins. The web as a graph: measurements, models, and methods. Proc. 5th Annual International Conference on Computing and Combinatorics (COCOON), 1–17 (1999).Google Scholar
  12. 12.
    S. R. Kumar, P. Raghavan, S. Rajagopalan and A. Tomkins. Trawling Emerging Cyber-Communities Automatically. Proc. 8th WWW Conference, Computer Networks, 31(11-16):1481–1493, (1999).CrossRefGoogle Scholar
  13. 13. E. L. Lawler. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston (1976).Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  1. 1.Stanford UniversityCAUSA

Personalised recommendations