An exact almost optimal algorithm for target set selection in social networks

The Target Set Selection problem proposed by Kempe, Kleinberg, and Tardos, gives a nice clean combinatorial for-mulation for many problems arising in economy, sociology, and medicine. Its input is a graph with vertex thresholds, the social network, and the goal is to find a subset of ver-tices, the target set, that “activates ” a prespecified number of vertices in the graph. Activation of a vertex is defined via a so-called activation process as follows: Initially, all ver-tices in the target set become active. Then at each step i of the process, each vertex gets activated if the number of its active neighbors at iteration i − 1 exceeds its threshold. The activation process is “monotone ” in the sense that once a vertex is activated, it remains active for the entire process. Unsurprisingly perhaps, Target Set Selection is NPC. More surprising is the fact that both of its maximiza-tion and minimization variants turn out to be extremely hard to approximate, even for very restrictive special cases. The only known case for which the problem is known to have some sort of acceptable worst-case solution is the case where the given social network is a tree and the problem be-comes polynomial-time solvable. In this paper, we attempt at extending this sparse landscape of tractable instances by considering the treewidth parameter of graphs. This param