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 \(k \ge 1\) 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 \(\frac{1}{10}\)-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 \(\frac{1}{2}\), when k is smaller than the number of vertices in the graph, and to \(\frac{2}{3}\), when k is a constant. For the computational complexity, we show that the problem is NP-hard even when \(k=3\).

图挖掘中的一个中心问题是找到密集的子图，在不同领域中有多种应用，其中一个著名的例子就是识别社区。尽管在查找单个密集子图的问题上已付出了很多努力，但直到最近才将重点转移到查找一组最密集子图的问题上。引入的查找可能重叠子图的方法是Top-k重叠最密集子图问题。给定一个整数\（k \ ge 1 \）和一个参数\（\ lambda> 0 \），此问题的目标是找到一组k个密集的子图，它们可以共享某些顶点。要最大化的目标函数考虑了子图的密度，参数\（\ lambda \）以及解决方案中每对子图之间的距离。的前k个重叠的子图最密集的问题已被证明承认一个\（\压裂{1} {10} \） -因子近似算法。此外，问题的计算复杂性尚未解决。在本文中，我们提出了有关问题的逼近性和计算复杂性的贡献。为了实现逼近，我们提出了一种近似算法，当k小于图中的顶点数量时，将逼近因子提高到\（\ frac {1} {2} \），并提高到\（\ frac {2} { 3} \），当k为常数时。对于计算复杂度，我们证明了即使在\（k = 3 \）。