On the Approximation of Special Instances of Minimum Topic-Connected Overlay

The design of a scalable overlay network to support decentralized topic-based publish/subscribe communication is nowadays a problem of great importance. We investigate here one such design problem called Minimum Topic-Connected Overlay. Given a collection of users together with the lists of topics they are interested in, connect these users to a network by a minimum number of edges such that every graph induced by users interested in one common topic is connected. It is known that this problem is APX-hard and approximable by a logarithmic factor. We focus here on hardness properties of some special instances. We study the problem where, for each topic, there are at most three users interested in it. Surprisingly, we show that even with such strong restriction, the problem stays NP-hard and it inherits the approximation hardness of the well-known vertex cover problem.