Inapproximability and parameterized results for the target set selection problem.

Given an undirected graph $G$ with a threshold function $\tau:V(G) \rightarrow \mathbb{N}$, the \emph{Target Set Selection (TSS) Problem} is about choosing a minimum cardinality set, say $S \subseteq V(G)$, such that starting a diffusion process with $S$ as its seed set will eventually result in activating all the nodes in $G$. Under the non-progressive diffusion model, we have the following results on the TSS Problem: We show that the TSS Problem on bipartite graphs does not admit an approximation algorithm with a performance guarantee asymptotically better than $O(\log n_{min})$, where $n_{min}$ is the cardinality of the smaller bipartition, unless $P=NP$. Though Chen et al. [SIDMA, 2009] had shown an inapproximability result of $\Omega(2^{\log^{1 - \epsilon} n})$ for the TSS Problem on general graphs under the progressive diffusion model, to the best of our knowledge, our result is the first inapproximability result under the non-progressive diffusion setting. Assuming a non-progressive diffusion model, it was shown by Nichterlein et al. [Social Network Analysis and Mining, 2013] that it is possible to compute an optimal sized target set in $O(2^{(2^{t}+1)t}\cdot m)$ time, where $m$ and $t$ denote the number of edges and the cardinality of a minimum vertex cover, respectively, of the graph under consideration. We improve this result by designing an algorithm that computes an optimal sized target set in $2^{O(t\log t)}n^{O(1)}$ time, where $n$ denotes the number of vertices of the graph under consideration.