# 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).

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