Abstract
We give an exact \(O(nk^2)\) algorithm for finding the densest k subgraph in outerplanar graphs. We extend this to an exact \(O(nk^2 8^b)\) algorithm for finding the densest k subgraph in b-outerplanar graphs. Often, when there is an exact polynomial time algorithm for a problem on b-outerplanar graphs, this algorithm can be extended to a polynomial time approximation scheme (PTAS) on planar graphs using Baker’s technique. We hypothesize that this is not possible for the densest k subgraph problem.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Angel, A., Sarkas, N., Koudas, N., Srivastava, D.: Dense subgraph maintenance under streaming edge weight updates for real-time story identification. Proc. VLDB Endow. 5(6), 574–585 (2012)
Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 153–180 (1994)
Bourgeois, N., Giannakos, A., Lucarelli, G., Milis, I., Paschos, V.T.: Exact and approximation algorithms for densest k-subgraph. In: WALCOM: Algorithms and Computation, pp. 114–125. Springer, Heidelberg (2013)
Buehrer, G., Chellapilla, K.: A scalable pattern mining approach to web graph compression with communities. In: Proceedings of the 2008 International Conference on Web Search and Data Mining, WSDM ’08, pp. 95–106. ACM, New York, NY, USA (2008)
Chen, J., Saad, Y.: Dense subgraph extraction with application to community detection. IEEE Trans. Knowl. Data Eng. 24(7), 1216–1230 (2010)
Corneil, D.G., Perl, Y.: Clustering and domination in perfect graphs. Discret. Appl. Math. 9(1), 27–39 (1984)
Du, X., Jin, R., Ding, L., Lee, V.E., Thornton Jr, J.H.: Migration motif: a spatial - temporal pattern mining approach for financial markets. In: Proceedings of the 15th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, KDD ’09, pp. 1135–1144. ACM, New York, NY, USA (2009)
Feige, U., Kortsarz, G., Peleg, D.: The dense k-subgraph problem. Algorithmica 29, 2001 (1999)
Fratkin, E., Naughton, B.T., Brutlag, D.L., Batzoglou, S.: MotifCut: regulatory motifs finding with maximum density subgraphs. Bioinformatics 22, 150–157 (2006)
Gibson, D., Kleinberg, J., Raghavan, P.: Inferring web communities from link topology. In: Proceedings of the Ninth ACM Conference on Hypertext and Hypermedia : Links, Objects, Time and Space—Structure in Hypermedia Systems: Links, Objects, Time and Space—Structure in Hypermedia Systems, HYPERTEXT ’98, pp. 225–234. ACM, New York, NY, USA (1998)
Gibson, D., Kumar, R., Tomkins, A.: Discovering large dense subgraphs in massive graphs. In: Proceedings of the 31st International Conference on Very Large Data Bases, VLDB ’05, pp. 721–732. VLDB Endowment (2005)
Goldberg, A.V.: Finding a maximum density subgraph. Technical report, University of California at Berkeley, Berkeley, CA, USA (1984)
Mark Keil, J., Brecht, T.B.: The complexity of clustering in planar graphs. J. Comb. Math. Comb. Comput. 9, 155–159 (1991)
Langston, M.A., Lin, L., Peng, X., Baldwin, N.E., Symons, C.T., Zhang, B., Snoddy, J.R.: A combinatorial approach to the analysis of differential gene expression data: the use of graph algorithms for disease prediction and screening. In: Methods of Microarray Data Analysis IV, pp. 223–238. Springer (2005)
Newman, M.E.J.: Fast algorithm for detecting community structure in networks. Phys. Rev. E 69, 066133 (2004)
Acknowledgements
We would like to express our sincere thanks to Samuel Chase for his collaboration on our initial explorations of finding a PTAS for the densest k subgraph problem. We would like to thank our reviewer who pointed us to previous work on this problem [3].
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
A Pseudocode Selections
A Pseudocode Selections
The merge procedure.
The extend procedure.
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Gonzales, S., Migler, T. (2020). The Densest k Subgraph Problem in b-Outerplanar Graphs. In: Cherifi, H., Gaito, S., Mendes, J., Moro, E., Rocha, L. (eds) Complex Networks and Their Applications VIII. COMPLEX NETWORKS 2019. Studies in Computational Intelligence, vol 881. Springer, Cham. https://doi.org/10.1007/978-3-030-36687-2_10
Download citation
DOI: https://doi.org/10.1007/978-3-030-36687-2_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-36686-5
Online ISBN: 978-3-030-36687-2
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)