Abstract
We consider the problem of finding dense subgraphs with specified upper or lower bounds on the number of vertices. We introduce two optimization problems: the densest at-least-k-subgraph problem (dalks), which is to find an induced subgraph of highest average degree among all subgraphs with at least k vertices, and the densest at-most-k-subgraph problem (damks), which is defined similarly. These problems are relaxed versions of the well-known densest k-subgraph problem (dks), which is to find the densest subgraph with exactly k vertices. Our main result is that dalks can be approximated efficiently, even for web-scale graphs. We give a (1/3)-approximation algorithm for dalks that is based on the core decomposition of a graph, and that runs in time O(m + n), where n is the number of nodes and m is the number of edges. In contrast, we show that damks is nearly as hard to approximate as the densest k-subgraph problem, for which no good approximation algorithm is known. In particular, we show that if there exists a polynomial time approximation algorithm for damks with approximation ratio γ, then there is a polynomial time approximation algorithm for dks with approximation ratio γ 2/8. In the experimental section, we test the algorithm for dalks on large publicly available web graphs. We observe that, in addition to producing near-optimal solutions for dalks, the algorithm also produces near-optimal solutions for dks for nearly all values of k.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Abello, J., Resende, M.G.C., Sudarsky, R.: Massive quasi-clique detection. In: Rajsbaum, S. (ed.) LATIN 2002. LNCS, vol. 2286, pp. 598–612. Springer, Heidelberg (2002)
Alvarez-Hamelin, J.I., Dall’Asta, L., Barrat, A., Vespignani, A.: Large scale networks fingerprinting and visualization using the k-core decomposition. Advances in Neural Information Processing Systems 18, 41–50 (2006)
Andersen, R.: A local algorithm for finding dense subgraphs. In: Proc. 19th ACM-SIAM Symposium on Discrete Algorithms (SODA 2008), pp. 1003–1009 (2008)
Arora, S., Karger, D., Karpinski, M.: Polynomial time approximation schemes for dense instances of NP-hard problems. In: Proc. 27th ACM Symposium on Theory of Computing (STOC 1995), pp. 284–293 (1995)
Asahiro, Y., Hassin, R., Iwama, K.: Complexity of finding dense subgraphs. Discrete Appl. Math. 121(1-3), 15–26 (2002)
Asahiro, Y., Iwama, K., Tamaki, H., Tokuyama, T.: Greedily finding a dense subgraph. J. Algorithms 34(2), 203–221 (2000)
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)
Buehrer, G., Chellapilla, K.: A scalable pattern mining approach to web graph compression with communities. In: WSDM 2008: Proceedings of the international conference on web search and web data mining, pp. 95–106 (2008)
Dourisboure, Y., Geraci, F., Pellegrini, M.: Extraction and classification of dense communities in the web. In: WWW 2007: Proceedings of the 16th international conference on World Wide Web, pp. 461–470 (2007)
Feige, U., Langberg, M.: Approximation algorithms for maximization problems arising in graph partitioning. J. Algorithms 41(2), 174–211 (2001)
Feige, U., Peleg, D., Kortsarz, G.: The dense k-subgraph problem. Algorithmica 29(3), 410–421 (2001)
Feige, U., Seltser, M.: On the densest k-subgraph problem, Technical report, Department of Applied Mathematics and Computer Science, The Weizmann Institute, Rehobot (1997)
Gallo, G., Grigoriadis, M., Tarjan, R.: A fast parametric maximum flow algorithm and applications. SIAM J. Comput. 18(1), 30–55 (1989)
Gibson, D., Kumar, R., Tomkins, A.: Discovering large dense subgraphs in massive graphs. In: Proc. 31st VLDB Conference (2005)
Goldberg, A.: Finding a maximum density subgraph, Technical Report UCB/CSB 84/171, Department of Electrical Engineering and Computer Science, University of California, Berkeley, CA (1984)
Kannan, R., Vinay, V.: Analyzing the structure of large graphs (manuscript) (1999)
Khot, S.: Ruling out PTAS for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM Journal on Computing 36(4), 1025–1071 (2006)
Kortsarz, G., Peleg, D.: Generating sparse 2-spanners. J. Algorithms 17(2), 222–236 (1994)
Seidman, S.B.: Network structure and minimum degree. Social Networks 5, 269–287 (1983)
Kumar, R., Raghavan, P., Rajagopalan, S., Tomkins, A.: Trawling the Web for emerging cyber-communities. In: Proc. 8th WWW Conference (WWW 1999) (1999)
Wuchty, S., Almaas, E.: Peeling the yeast protein network. Proteomics 5, 444 (2005)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Andersen, R., Chellapilla, K. (2009). Finding Dense Subgraphs with Size Bounds. In: Avrachenkov, K., Donato, D., Litvak, N. (eds) Algorithms and Models for the Web-Graph. WAW 2009. Lecture Notes in Computer Science, vol 5427. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-95995-3_3
Download citation
DOI: https://doi.org/10.1007/978-3-540-95995-3_3
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-95994-6
Online ISBN: 978-3-540-95995-3
eBook Packages: Computer ScienceComputer Science (R0)