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.
KeywordsDirected Graph Greedy Algorithm Undirected Graph Dual Solution Dense Subgraph
Unable to display preview. Download preview PDF.
- 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
- 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.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.U. Feige and M. Seltser. On the Densest k-Subgraph Problem. Weizmann Institute Technical Report CS 97-16 (1997).Google Scholar
- 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
- 8.D. Gibson, J. Kleinberg and P. Raghavan. Inferring web communities from Web topology. Proc. HYPERTEXT, 225–234 (1998).Google Scholar
- 9.R. Kannan and V. Vinay. Analyzing the Structure of Large Graphs. manuscript, August 1999.Google Scholar
- 10.J. Kleinberg. Authoritative sources in hypertext linked environments. Proc. 9th Annual ACM-SIAM Symposium on Discrete Algorithms, 668–677 (1998).Google Scholar
- 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
- 13. E. L. Lawler. Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston (1976).Google Scholar