Parameterized complexity of multi-node hubs

Hubs are high-degree nodes within a network. The examination of the emergence and centrality of hubs lies at the heart of many studies of complex networks such as telecommunication networks, biological networks, social networks and semantic networks. Furthermore, identifying and allocating hubs are routine tasks in applications. In this paper, we do not seek a hub that is a single node, but a hub that consists of k nodes. Formally, given a graph G = (V, E), we a seek a set A ? V of size k that induces a connected subgraph from which at least p edges emanate. Thus, we identify k nodes which can act as a unit (due to the connectivity constraint) that is a hub (due to the cut constraint). This problem, which we call Multi-Node Hub (MNH), can also be viewed as a variant of the classic Max Cut problem. While it is easy to see that MNH is W[1]-hard with respect to the parameter k, our main contribution is the first parameterized algorithm that shows that MNH is FPT with respect to the parameter p. Despite recent breakthrough advances for cut-problems like Multicut and Minimum Bisection, MNH is still very challenging. Not only does a connectivity constraint has to be handled on top of the involved machinery developed for these problems, but also the fact that MNH is a maximization problem seems to prevent the applicability of this machinery in the first place. To deal with the latter issue, we give non-trivial reduction rules that show how MNH can be preprocessed into a problem where it is necessary to delete a bounded-in-parameter number of vertices. Then, to handle the connectivity constraint, we use a novel application of the form of tree decomposition introduced by Cygan et al. [STOC 2014] to solve Minimum Bisection, where we demonstrate how connectivity constraints can be replaced by simpler size constraints. Our approach may be relevant to the design of algorithms for other cut-problems of this nature.