The Densest Subgraph Problem with a Convex/Concave Size Function

Abstract

In the densest subgraph problem, given an edge-weighted undirected graph G = (V, E, w), we are asked to find S ⊆ 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 ⊆ V . Specifically, we introduce the f -density of S ⊆ V , which is defined as w(S)/f (|S|), where f : Z ≥0 → R ≥0 is a monotonically non-decreasing function. In the f -densest subgraph problem (f -DS), we aim to find S ⊆ 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.

In Search of the Densest Subgraph

“…A different generalization of the edge density was proposed by Kawase and Miyauchi [12]. Let f be a positive real-valued, non-decreasing function.…”

“…Thus, indeed, the larger of the equally dense graphs is preferred. An important general result of Kawase and Miyauchi [12] is that the f -densest subgraph can be found in polynomial time for any real-valued, non-decreasing, polynomial-time computable concave function. On the other hand, if the function is strictly convex, then the problem becomes NP-hard.…”

Graph Mining Meets Crowdsourcing: Extracting Experts for Answer Aggregation

“…. , S |V | } that maximizes d Si (v i ), although there are other variants depending on the problem at hand [7,17,30,33]. Algorithm 1 can be implemented to run in O(|E| + |V | log |V |) time.…”

Robust Densest Subgraph Discovery

“…In addition to the variant on uncertain graphs, the densest subgraph problem has numerous noteworthy problem variations. Examples include the size-constraint variants [1], [10], [17], [27], [34] and the variants generalizing the term w(S) in the density [30], [32], [35] and the term |S| in the density [26]. Furthermore, a large body of work has been devoted to the streaming or dynamic settings of the densest subgraph problem [5], [11], [16], [24], [29], [33].…”