Computing the k densest subgraphs of a graph

Computing cohesive subgraphs is a central problem in graph theory. While many formulations of cohesive subgraphs lead to NP-hard problems, finding a densest subgraph can be done in polynomial-time. As such, the densest subgraph model has emerged as the most popular notion of cohesiveness. Recently, the data mining community has started looking into the problem of computing k densest subgraphs in a given graph, rather than one. In this paper we consider a natural variant of the k densest subgraphs problem, where overlap between solution subgraphs is allowed with no constraint. We show that the problem is fixed-parameter tractable with respect to k, and admits a PTAS for constant k. Both these algorithms complement nicely the previously known O (nk) algorithm for the problem. (c) 2022 Elsevier B.V. All rights reserved.