Top-k overlapping densest subgraphs: approximation algorithms and computational 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 lambda > 0, the goal of this problem is to find a set of k dense subgraphs that may share some vertices. The objective function to be maximized takes into account the density of the subgraphs, the parameter lambda and the distance between each pair of subgraphs in the solution. The Top-k-Overlapping Densest Subgraphs problem has been shown to admit a 1/10-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 1/2, 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.

Automated summary

The central problem in graph mining is the discovery of dense subgraphs, with a recent focus on finding a set of densest subgraphs. The Top-k-Overlapping Densest Subgraphs problem aims to find a set of k dense subgraphs that may share vertices, with an objective function considering density, parameter lambda, and distance. Research has shown a 1/10-factor approximation algorithm for this problem, while also proving its NP-hardness.