Abstract
The density of a subgraph in an undirected graph is the sum of the subgraph’s edge weights divided by the number of the subgraph’s vertices. Finding an induced subgraph of maximum density among all subgraphs with at least k vertices is called as the densest at-least-k-subgraph problem (DalkS).
In this paper, we first present a polynomial time algorithms for DalkS when k is bounded by some constant c. For a graph of n vertices and m edges, our algorithm is of time complexity \(O(n^{c + 3} \log n)\), which improve previous best time complexity \(O(n^c(n+m)^{4.5})\).
Second, we give a greedy approximation algorithm for the Densest Subgraph with a Specified Subset Problem. We show that the greedy algorithm is of approximation ratio \(2\cdot (1+ \frac{k}{3})\), where k is the element number of the specified subset.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
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)
Chen, W., Samatova, N.F., Stallmann, M.F., Hendrix, W.: On size-constrained minimum \(s\)-\(t\) cut problems and size-constrained dense subgraph problems, submitted to Theoretical Computer Science, under review
Feige, U., Seltser, M.: On the densest k-subgraph problems. Technical report: CS97-16, Department of Applied Mathematics and Computer Science (1997)
Feige, U., Kortsarz, G., Peleg, D.: The dense \(k\)-subgraph problem. Algorithmica 29, 410–421 (2001)
Gallo, G., Grigoriadis, M., Tarjan, R.: A fast parametric maximum flow algorithm and applications. SIAM J. Comput. 18(1), 30–55 (1989)
Goldberg, A.: Finding a maximum density subgraph, Technical report UCB/CSB 84/171, Department of Electrical Engineering and Computer Science, University of California, Berkeley (1984)
Hu, H., Yan, X., Huang, Y., et al.: Mining coherent dense subgraphs across massive biological networks for functional discovery. Bioinformatics 21, 213–221 (2005)
Han, Q.M., Ye, Y.Y., Zhang, J.W.: Approximation of Dense-\(k\) subgraph, http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.41.1899
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)
Kortsarz, G., Peleg, D.: On choosing a dense subgraph. In: Proceedings of the 34th Annual IEEE Symposium on Foundations of Computer Science, pp. 692–701 (1993)
Megiddo, N.: Combinatorial optimization with rational objective function. Math. Operat. Res. 4(4), 414–424 (1979)
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)
Srivastav, A., Wolf, K.: Finding dense subgraphs with semidefinite programming. In: Jansen, K., Rolim, J.D.P. (eds.) APPROX 1998. LNCS, vol. 1444, pp. 181–191. Springer, Heidelberg (1998)
Acknowledgments
We would like to thank the anonymous referees for their careful readings of the manuscripts and many useful suggestions.sparabreak Wenbin Chen’s research has been supported by the National Science Foundation of China (NSFC) under Grant No. 11271097. Lingxi Peng’s research has been partly supported by the Funding Program for Research Development in Institutions of Higher Learning Under the Jurisdiction of Guangzhou Municipality under Grant No. 2012A077. Jianxiong Wang’s research was partially supported under Foundation for Distinguished Young Talents in Higher Education of Guangdong (2012WYM0105 and 2012LYM0105) and Funding Program for Research Development in Institutions of Higher Learning Under the Jurisdiction of Guangzhou Municipality (2012A143). FuFang Li’s work had been co-financed by: Natural Science Foundation of China under Grant No. 61472092; Guangdong Provincial Science and Technology Plan Project under Grant No. 2013B010401037; and GuangZhou Municipal High School Science Research Fund under grant No. 1201421317. Maobin Tang’s research has been supported under Guangdong Province’s Science and Technology Projects under Grant No. 2012A020602065 and the research project of Guangzhou education bureau under Grant No. 2012A075.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Chen, W., Peng, L., Wang, J., Li, F., Tang, M. (2015). Algorithms for the Densest Subgraph with at Least k Vertices and with a Specified Subset. In: Lu, Z., Kim, D., Wu, W., Li, W., Du, DZ. (eds) Combinatorial Optimization and Applications. Lecture Notes in Computer Science(), vol 9486. Springer, Cham. https://doi.org/10.1007/978-3-319-26626-8_41
Download citation
DOI: https://doi.org/10.1007/978-3-319-26626-8_41
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-26625-1
Online ISBN: 978-3-319-26626-8
eBook Packages: Computer ScienceComputer Science (R0)