# The Densest Subgraph Problem with a Convex/Concave Size Function

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## Abstract

In the densest subgraph problem, given an edge-weighted undirected graph \(G=(V,E,w)\), we are asked to find \(S\subseteq V\) that maximizes the *density*, i.e., *w*(*S*) / |*S*|, where *w*(*S*) is the sum of weights of the edges in the subgraph induced by *S*. This problem has often been employed in a wide variety of graph mining applications. However, the problem has a drawback; it may happen that the obtained subset is too large or too small in comparison with the size desired in the application at hand. In this study, we address the size issue of the densest subgraph problem 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 \(f:\mathbb {Z}_{\ge 0}\rightarrow \mathbb {R}_{\ge 0}\) is a monotonically non-decreasing function. In the *f*-*densest subgraph problem* (*f*-DS), we aim 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*/*concave* size function *f* appropriately. For *f*-DS with convex function *f*, we propose a nearly-linear-time algorithm with a provable approximation guarantee. On the other hand, for *f*-DS with concave function *f*, we propose an LP-based exact algorithm, a flow-based \(O(|V|^3)\)-time exact algorithm for unweighted graphs, and a nearly-linear-time approximation algorithm.

## Keywords

Graphs Dense subgraph extraction Densest subgraph problem Approximation algorithms## Notes

### Acknowledgements

The authors would like to thank the anonymous reviewers for their valuable suggestions and helpful comments. The authors would also like to thank Yoshio Okamoto and Akiko Takeda for pointing out the references [20] and [18], respectively. The first author is supported by a Grant-in-Aid for Young Scientists (B) (No. 16K16005). The second author was supported by a Grant-in-Aid for JSPS Fellows (No. 26-11908), and is supported by a Grant-in-Aid for Research Activity Start-up (No. 17H07357).

## References

- 1.Andersen, R., Chellapilla, K.: Finding dense subgraphs with size bounds. In: WAW ’09: Proceedings of the 6th Workshop on Algorithms and Models for the Web Graph, pp. 25–37 (2009)CrossRefGoogle Scholar
- 2.Angel, A., Sarkas, N., Koudas, N., Srivastava, D.: Dense subgraph maintenance under streaming edge weight updates for real-time story identification. In: VLDB ’12: Proceedings of the 38th International Conference on Very Large Data Bases, pp. 574–585 (2012)CrossRefGoogle Scholar
- 3.Asahiro, Y., Iwama, K., Tamaki, H., Tokuyama, T.: Greedily finding a dense subgraph. J. Algorithms
**34**(2), 203–221 (2000)MathSciNetCrossRefGoogle Scholar - 4.Bader, G.D., Hogue, C.W.V.: An automated method for finding molecular complexes in large protein interaction networks. BMC Bioinform.
**4**(1), 1–27 (2003)CrossRefGoogle Scholar - 5.Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting high log-densities: an \({O}(n^{1/4})\) approximation for densest \(k\)-subgraph. In: STOC ’10: Proceedings of the 42nd ACM Symposium on Theory of Computing, pp. 201–210 (2010)Google Scholar
- 6.Bonchi, F., Gullo, F., Kaltenbrunner, A., Volkovich, Y.: Core decomposition of uncertain graphs. In: KDD ’14: Proceedings of the 20th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 1316–1325 (2014)Google Scholar
- 7.Charikar, M.: Greedy approximation algorithms for finding dense components in a graph. In: APPROX ’00: Proceedings of the 3rd International Workshop on Approximation Algorithms for Combinatorial Optimization, pp. 84–95 (2000)CrossRefGoogle Scholar
- 8.Cheriyan, J., Hagerup, T., Mehlhorn, K.: An \(o(n^3)\)-time maximum-flow algorithm. SIAM J. Comput.
**25**(6), 1144–1170 (1996)MathSciNetCrossRefGoogle Scholar - 9.Dourisboure, Y., Geraci, F., Pellegrini, M.: Extraction and classification of dense communities in the web. In: WWW ’07: Proceedings of the 16th International Conference on World Wide Web, pp. 461–470 (2007)Google Scholar
- 10.Feige, U., Peleg, D., Kortsarz, G.: The dense \(k\)-subgraph problem. Algorithmica
**29**(3), 410–421 (2001)MathSciNetCrossRefGoogle Scholar - 11.Fratkin, E., Naughton, B.T., Brutlag, D.L., Batzoglou, S.: MotifCut: regulatory motifs finding with maximum density subgraphs. Bioinformatics
**22**(14), e150–e157 (2006)CrossRefGoogle Scholar - 12.Fujishige, S.: Submodular Functions and Optimization, Annals of Discrete Mathematics, vol. 58. Elsevier, Amsterdam (2005)zbMATHGoogle Scholar
- 13.Gibson, D., Kumar, R., Tomkins, A.: Discovering large dense subgraphs in massive graphs. In: VLDB ’05: Proceedings of the 31st International Conference on Very Large Data Bases, pp. 721–732 (2005)Google Scholar
- 14.Goldberg, A.V.: Finding a maximum density subgraph. Technical report, University of California Berkeley (1984)Google Scholar
- 15.Kawase, Y., Miyauchi, A.: The densest subgraph problem with a convex/concave size function. In: ISAAC ’16: Proceedings of the 27th International Symposium on Algorithms and Computation, pp. 44:1–44:12 (2016)Google Scholar
- 16.Khot, S.: Ruling out PTAS for graph min-bisection, dense \(k\)-subgraph, and bipartite clique. SIAM J. Comput.
**36**(4), 1025–1071 (2006)MathSciNetCrossRefGoogle Scholar - 17.Khuller, S., Saha, B.: On finding dense subgraphs. In: ICALP ’09: Proceedings of the 36th International Colloquium on Automata, Languages and Programming, pp. 597–608 (2009)CrossRefGoogle Scholar
- 18.Lee, Y.T., Sidford, A., Wong, S.C.W.: A faster cutting plane method and its implications for combinatorial and convex optimization. In: FOCS ’15: Proceedings of the 56th Annual IEEE Symposium on Foundations of Computer Science, pp. 1049–1065 (2015)Google Scholar
- 19.Megiddo, N.: Combinatorial optimization with rational objective functions. Math. Oper. Res.
**4**(4), 414–424 (1979)MathSciNetCrossRefGoogle Scholar - 20.Nagano, K., Kawahara, Y., Aihara, K.: Size-constrained submodular minimization through minimum norm base. In: ICML ’11: Proceedings of the 28th International Conference on Machine Learning, pp. 977–984 (2011)Google Scholar
- 21.Tsourakakis, C.E., Bonchi, F., Gionis, A., Gullo, F., Tsiarli, M.: Denser than the densest subgraph: extracting optimal quasi-cliques with quality guarantees. In: KDD ’13: Proceedings of the 19th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 104–112 (2013)Google Scholar
- 22.Yanagisawa, H., Hara, S.: Discounted average degree density metric and new algorithms for the densest subgraph problem. Networks. https://doi.org/10.1002/net.21764 (in press)MathSciNetCrossRefGoogle Scholar