Latency-Bounded Target Set Selection in Social Networks

Motivated by applications in sociology, economy and medicine, we study variants of the Target Set Selection problem, first proposed by Kempe, Kleinberg and Tardos. In our scenario one is given a graph \(G=(V,E)\), integer values \(t(v)\) for each vertex \(v\) (\emph{thresholds}), and the objective is to determine a small set of vertices (\emph{target set}) that activates a given number (or a given subset) of vertices of \(G\) \emph{within} a prescribed number of rounds. The activation process in \(G\) proceeds as follows: initially, at round 0, all vertices in the target set are activated; subsequently at each round \(r\geq 1\) every vertex of \(G\) becomes activated if at least \(t(v)\) of its neighbors are already active by round \(r-1\). It is known that the problem of finding a minimum cardinality Target Set that eventually activates the whole graph \(G\) is hard to approximate to a factor better than \(O(2^{\log^{1-\epsilon}|V|})\). In this paper we give \emph{exact} polynomial time algorithms to find minimum cardinality Target Sets in graphs of bounded clique-width, and \emph{exact} linear time algorithms for trees.