Influence is a Matter of Degree: New Algorithms for Activation Problems

We consider the target set selection problem. In this problem, a vertex is active either if it belongs to a set of initially activated vertices or if at some point it has at least $k$ active neighbors ($k$ is identical for all vertices of the graph). Our goal is to find a set of minimum size whose activation will result with the entire graph being activated. Call such a set \emph{contagious}. We prove that if $G=(V,E)$ is an undirected graph, the size of a contagious set is bounded by $\sum_{v\in V}{\min \{1,\frac{k}{d(v)+1}\}}$ (where $d(v)$ is the degree of $v$). We present a simple and efficient algorithm that finds a contagious set that is not larger than the aforementioned bound and discuss algorithmic applications of this algorithm to finding contagious sets in dense graphs.