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 $f:\mathbb{Z}_{\geq 0}\rightarrow \mathbb{R}_{\geq 0}$ 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.
Highlights
Finding dense components in a graph is an active research topic in graph mining
Let S∗ ⊆ V be an optimal solution to f -densest subgraph problem (f -DS) and SD∗ S ⊆ V be the densest subgraph
Let S∗ ⊆ V be an optimal solution to f -DS with convex function f
Summary
Finding dense components in a graph is an active research topic in graph mining. Techniques for identifying dense subgraphs have been used in various applications. In the (weighted) densest subgraph problem, given an (edge-weighted) undirected graph G = (V, E, w), we are asked to find S ⊆ V that maximizes the density w(S)/|S|. The first problem, the densest at-least-k-subgraph problem (DalkS), asks for S ⊆ V that maximizes the density w(S)/|S| under the size constraint |S| ≥ k. For this problem, Andersen and Chellapilla [1] adopted the greedy peeling, and demonstrated that the algorithm yields a 3-approximate solution for any instance. Khuller and Saha [15] investigated the problem more deeply They proved that DalkS is NP-hard, and designed a flow-based algorithm and an LP-based algorithm. Khuller and Saha [15] proved that approximating DamkS is as hard as approximating DkS within a constant factor
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