Fast Algorithms for Constrained Graph Density Problems

  • Venkatesan Chakaravarthy
  • Neelima Gupta
  • Aditya Pancholi
  • Sambuddha Roy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8973)


We consider the question of finding communities in large social networks. In literature and practice, “communities” refer to a well-connected subgraph of the entire network. For instance, the notion of graph density has been considered as a reasonable measure of a community. Researchers have also looked at the minimum degree of a subgraph as a measure of the connectedness of the community.

Typically, a community is meaningful in the context of a social network if it is of somewhat significant size. Thus, earlier work has considered the densest graph problem subject to various co-matroid constraints. Most of these algorithms utilize an exact dense subgraph procedure as a subroutine; such a subroutine involves computing maximum flows or solving LPs. Consequently, they are rather inefficient when considered for massive graphs. For massive graphs, we are constrained to run in near-linear time, while producing subgraphs that provide reasonable approximations to the optimal solutions.

Our current work presents efficient greedy algorithms for the problem of graph density subject to an even more general class of constraints called upward-monotone constraints (these subsume co-matroid constraints). This generalizes and extends earlier work significantly. For instance, we are thereby able to present near-linear time 3-factor approximation algorithms for density subject to co-matroid constraints; we are also able to obtain 2-factor LP-based algorithms for density subject to 2 co-matroid constraints.

Our algorithms heavily utilize the core decomposition of a graph.


Minimum Degree Vertex Cover Unconstrained Optimum Problem Extend Graph Density 
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.
    Alon, N., Spencer, J.H.: The Probabilistic Method. Wiley, New York (1992)zbMATHGoogle Scholar
  2. 2.
    Andersen, R., Chellapilla, K.: Finding dense subgraphs with size bounds. In: Avrachenkov, K., Donato, D., Litvak, N. (eds.) WAW 2009. LNCS, vol. 5427, pp. 25–37. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting high log-densities: an o(n1/4) approximation for densest k-subgraph. In: STOC, pp. 201–210 (2010)Google Scholar
  4. 4.
    Chakaravarthy, V.T., Modani, N., Natarajan, S.R., Roy, S., Sabharwal, Y.: Density functions subject to a co-matroid constraint. In: FSTTCS, pp. 236–248 (2012)Google Scholar
  5. 5.
    Charikar, M.: Greedy approximation algorithms for finding dense components in a graph. In: Jansen, K., Khuller, S. (eds.) APPROX 2000. LNCS, vol. 1913, pp. 84–95. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  6. 6.
    Eisenstat, D., Klein, P.N.: Linear-time algorithms for max flow and multiple-source shortest paths in unit-weight planar graphs. In: Symposium on Theory of Computing Conference, STOC 2013, Palo Alto, CA, USA, June 1-4, pp. 735–744 (2013)Google Scholar
  7. 7.
    Feige, U., Kortsarz, G., Peleg, D.: The dense k-subgraph problem. Algorithmica 29, 2001 (1999)MathSciNetGoogle Scholar
  8. 8.
    Gajewar, A., Sarma, A.D.: Multi-skill collaborative teams based on densest subgraphs. In: SDM, pp. 165–176 (2012)Google Scholar
  9. 9.
    Goldberg, A.V.: Finding a maximum density subgraph. Technical report, UC Berkeley (1984)Google Scholar
  10. 10.
    Kelner, J.A., Lee, Y.T., Orecchia, L., Sidford, A.: An almost-linear-time algorithm for approximate max flow in undirected graphs, and its multicommodity generalizations. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, pp. 217–226 (2014)Google Scholar
  11. 11.
    Khot, S.: Ruling Out PTAS for Graph Min-Bisection, Dense k-Subgraph, and Bipartite Clique. SIAM J. Comput. 36(4), 1025–1071 (2006)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Khuller, S., Saha, B.: On finding dense subgraphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009, Part I. LNCS, vol. 5555, pp. 597–608. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  13. 13.
    Kortsarz, G., Peleg, D.: Generating sparse 2-spanners. J. Algorithms 17(2), 222–236 (1994)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Lawler, E.: Combinatorial optimization - networks and matroids. Holt, Rinehart and Winston, New York (1976)zbMATHGoogle Scholar
  15. 15.
    Matula, D.W., Beck, L.L.: Smallest-last ordering and clustering and graph coloring algorithms. J. ACM 30(3), 417–427 (1983)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Rangapuram, S.S., Bühler, T., Hein, M.: Towards realistic team formation in social networks based on densest subgraphs. In: Proceedings of the 22nd International Conference on World Wide Web, WWW 2013, pp. 1077–1088 (2013)Google Scholar
  17. 17.
    Saha, B., Hoch, A., Khuller, S., Raschid, L., Zhang, X.-N.: Dense subgraphs with restrictions and applications to gene annotation graphs. In: Berger, B. (ed.) RECOMB 2010. LNCS, vol. 6044, pp. 456–472. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  18. 18.
    Schrijver, A.: Combinatorial Optimization - Polyhedra and Efficiency. Springer (2003)Google Scholar
  19. 19.
    Sherman, J.: Nearly maximum flows in nearly linear time. In: 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, Berkeley, CA, USA, October 26-29, pp. 263–269 (2013)Google Scholar
  20. 20.
    Sozio, M., Gionis, A.: The community-search problem and how to plan a successful cocktail party. In: KDD, pp. 939–948 (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Venkatesan Chakaravarthy
    • 1
  • Neelima Gupta
    • 2
  • Aditya Pancholi
    • 2
  • Sambuddha Roy
    • 3
  1. 1.IBM ResearchBangaloreIndia
  2. 2.University of DelhiDelhiIndia
  3. 3.AmazonBangaloreIndia

Personalised recommendations