Greedy approximation algorithms for finding dense components in a graph

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.

[1]  Jon M. Kleinberg,et al.  Inferring Web communities from link topology , 1998, HYPERTEXT '98.

[2]  Robert E. Tarjan,et al.  A Fast Parametric Maximum Flow Algorithm and Applications , 1989, SIAM J. Comput..

[3]  U. Feige,et al.  On the densest k-subgraph problems , 1997 .

[4]  Alan M. Frieze,et al.  Fast Monte-Carlo algorithms for finding low-rank approximations , 1998, Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280).

[5]  Jon Kleinberg,et al.  Authoritative sources in a hyperlinked environment , 1998, SODA '98.

[6]  Alan M. Frieze,et al.  Clustering in large graphs and matrices , 1999, SODA '99.

[7]  Ravi Kumar,et al.  Trawling the Web for Emerging Cyber-Communities , 1999, Comput. Networks.

[8]  Yuichi Asahiro,et al.  Finding Dense Subgraphs , 1995, ISAAC.

[9]  Uriel Feige,et al.  The Dense k -Subgraph Problem , 2001, Algorithmica.

[10]  Jon M. Kleinberg,et al.  The Web as a Graph: Measurements, Models, and Methods , 1999, COCOON.

[11]  Hisao Tamaki,et al.  Greedily Finding a Dense Subgraph , 2000, J. Algorithms.