Top-k overlapping densest subgraphs: Approximation and complexity

A central problem in graph mining is finding dense subgraphs, with several applications in different fields, a notable example being identifying communities. While a lot of effort has been put in the problem of finding a single dense subgraph, only recently the focus has been shifted to the problem of finding a set of densest subgraphs. An approach introduced to find possible overlapping subgraphs is the Top-k-Overlapping Densest Subgraphs problem. Given an integer k ≥ 1 and a parameter λ > 0, the goal of this problem is to find a set of k densest subgraphs that may share some vertices. The objective function to be maximized takes into account the density of the subgraphs and the distance between subgraphs in the solution (multiplied by λ). The Top-k-Overlapping Densest Subgraphs problem has been shown to admit a ⅒-factor approximation algorithm. Furthermore, the computational complexity of the problem has been left open. In this paper, we present contributions concerning the approximability and the computational complexity of the problem. For the approximability, we present approximation algorithms that improve the approximation factor to ½, when k is smaller than the number of vertices in the graph, and to 2/3, when k is a constant. For the computational complexity, we show that the problem is NP-hard even when k = 3.