Influence Diffusion in Social Networks under Time Window Constraints

We study a combinatorial model of the spread of influence in networks that generalizes existing schemata recently proposed in the literature. In our model, agents change behaviors/opinions on the basis of information collected from their neighbors in a time interval of bounded size whereas agents are assumed to have unbounded memory in previously studied scenarios. In our mathematical framework, one is given a network $G=(V,E)$, an integer value $t(v)$ for each node $v\in V$, and a time window size $\lambda$. The goal is to determine a small set of nodes (target set) that influences the whole graph. The spread of influence proceeds in rounds as follows: initially all nodes in the target set are influenced; subsequently, in each round, any uninfluenced node $v$ becomes influenced if the number of its neighbors that have been influenced in the previous $\lambda$ rounds is greater than or equal to $t(v)$. We prove that the problem of finding a minimum cardinality target set that influences the whole network $G$ is hard to approximate within a polylogarithmic factor. On the positive side, we design exact polynomial time algorithms for paths, rings, trees, and complete graphs.Comment: An extended abstract of a preliminary version of this paper appeared in: Proceedings of 20th International Colloquium on Structural Information and Communication Complexity (Sirocco 2013), Lectures Notes in Computer Science vol. 8179, T. Moscibroda and A.A. Rescigno (Eds.), pp. 141-152, 201

Tight Inapproximability of Target Set Reconfiguration

Given a graph $G$ with a vertex threshold function $\tau$, consider a dynamic process in which any inactive vertex $v$ becomes activated whenever at least $\tau(v)$ of its neighbors are activated. A vertex set $S$ is called a target set if all vertices of $G$ would be activated when initially activating vertices of $S$. In the Minmax Target Set Reconfiguration problem, for a graph $G$ and its two target sets $X$ and $Y$, we wish to transform $X$ into $Y$ by repeatedly adding or removing a single vertex, using only target sets of $G$, so as to minimize the maximum size of any intermediate target set. We prove that it is NP-hard to approximate Minmax Target Set Reconfiguration within a factor of $2-o\left(\frac{1}{\operatorname{polylog} n}\right)$, where $n$ is the number of vertices. Our result establishes a tight lower bound on approximability of Minmax Target Set Reconfiguration, which admits a $2$-factor approximation algorithm. The proof is based on a gap-preserving reduction from Target Set Selection to Minmax Target Set Reconfiguration, where NP-hardness of approximation for the former problem is proven by Chen (SIAM J. Discrete Math., 2009) and Charikar, Naamad, and Wirth (APPROX/RANDOM 2016).Comment: 13 page