The Densest Subgraph Problem with a Convex/Concave Size Function

Authors Yasushi Kawase, Atsushi Miyauchi

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Yasushi Kawase
Atsushi Miyauchi

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Yasushi Kawase and Atsushi Miyauchi. The Densest Subgraph Problem with a Convex/Concave Size Function. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 44:1-44:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)


Given an edge-weighted undirected graph G = (V, E, w), the density of S subseteq V is defined as w(S)/|S|, where w(S) is the sum of weights of the edges in the subgraph induced by S. The densest subgraph problem asks for S subseteq V that maximizes the density w(S)/|S|. The problem has received significant attention recently because it can be solved exactly in polynomial time. However, the densest subgraph problem has a drawback; it may happen that the obtained subset is too large or too small in comparison with the desired size of the output. In this study, we address the size issue by generalizing the density of S subseteq V. Specifically, we introduce the f -density of S subseteq V, which is defined as w(S)/f (|S|), where f : Z geq 0 to R geq 0 is a monotonically non-decreasing function. In the f-densest subgraph problem (f-DS), we are asked to find S subseteq V that maximizes the f-density w(S)/f (|S|). Although f-DS does not explicitly specify the size of the output subset of vertices, we can handle the above size issue using a convex size function f or a concave size function f appropriately. For f-DS with convex function f, we propose a nearly-linear-time algorithm with a provable approximation guarantee. In particular, for f-DS with f(x) = x^alpha (alpha in [1, 2]), our algorithm has an approximation ratio of 2 · n^{(alpha-1)(2-alpha)}. On the other hand, for f-DS with concave function f , we propose a linear-programming-based polynomial-time exact algorithm. It should be emphasized that this algorithm obtains not only an optimal solution to the problem but also subsets of vertices corresponding to the extreme points of the upper convex hull of {(|S|, w(S)) | S subseteq V }, which we refer to as the dense frontier points. We also propose a flow-based combinatorial exact algorithm for unweighted graphs that runs in O(n^3) time. Finally, we propose a nearly-linear-time 3-approximation algorithm.
  • graphs
  • dense subgraph extraction
  • densest subgraph problem
  • approxi- mation algorithms


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  1. R. Andersen and K. Chellapilla. Finding dense subgraphs with size bounds. In WAW'09, pages 25-37, 2009. Google Scholar
  2. A. Angel, N. Sarkas, N. Koudas, and D. Srivastava. Dense subgraph maintenance under streaming edge weight updates for real-time story identification. In VLDB'12, pages 574-585, 2012. Google Scholar
  3. Y. Asahiro, K. Iwama, H. Tamaki, and T. Tokuyama. Greedily finding a dense subgraph. J. Algorithms, 34(2):203-221, 2000. Google Scholar
  4. G. D. Bader and C. W. V. Hogue. An automated method for finding molecular complexes in large protein interaction networks. BMC Bioinformatics, 4(1):1-27, 2003. Google Scholar
  5. A. Bhaskara, M. Charikar, E. Chlamtac, U. Feige, and A. Vijayaraghavan. Detecting high log-densities: An O(n^1/4) approximation for densest k-subgraph. In STOC'10, pages 201-210, 2010. Google Scholar
  6. F. Bonchi, F. Gullo, A. Kaltenbrunner, and Y. Volkovich. Core decomposition of uncertain graphs. In KDD'14, pages 1316-1325, 2014. Google Scholar
  7. M. Charikar. Greedy approximation algorithms for finding dense components in a graph. In APPROX'00, pages 84-95, 2000. Google Scholar
  8. J. Cheriyan, T. Hagerup, and K. Mehlhorn. An o(n³)-time maximum-flow algorithm. SIAM J. Comput., 25(6):1144-1170, 1996. Google Scholar
  9. Y. Dourisboure, F. Geraci, and M. Pellegrini. Extraction and classification of dense communities in the web. In WWW'07, pages 461-470, 2007. Google Scholar
  10. E. Fratkin, B. T. Naughton, D. L. Brutlag, and S. Batzoglou. MotifCut: regulatory motifs finding with maximum density subgraphs. Bioinformatics, 22(14):e150-e157, 2006. Google Scholar
  11. S. Fujishige. Submodular Functions and Optimization, volume 58 of Annals of Discrete Mathematics. Elsevier, 2005. Google Scholar
  12. D. Gibson, R. Kumar, and A. Tomkins. Discovering large dense subgraphs in massive graphs. In VLDB'05, pages 721-732, 2005. Google Scholar
  13. A. V. Goldberg. Finding a maximum density subgraph. Technical report, University of California Berkeley, 1984. Google Scholar
  14. S. Khot. Ruling out PTAS for graph min-bisection, dense k-subgraph, and bipartite clique. SIAM J. Comput., 36(4):1025-1071, 2006. Google Scholar
  15. S. Khuller and B. Saha. On finding dense subgraphs. In ICALP'09, pages 597-608, 2009. Google Scholar
  16. J. B. Orlin. A faster strongly polynomial time algorithm for submodular function minimization. Math. Program., 118(2):237-251, 2009. Google Scholar
  17. C. E. Tsourakakis, F. Bonchi, A. Gionis, F. Gullo, and M. Tsiarli. Denser than the densest subgraph: Extracting optimal quasi-cliques with quality guarantees. In KDD'13, pages 104-112, 2013. Google Scholar
  18. H. Yanagisawa and S. Hara. Axioms of density: How to define and detect the densest subgraph. Technical report, IBM Research - Tokyo, 2016. Google Scholar
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