'Target Set Selection' on Graphs of Bounded Vertex Cover Number

Given a simple, undirected graph $G$ with a threshold function $\tau:V(G) \rightarrow \mathbb{N}$, the \textsc{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 $G$. We have the following results on the TSS Problem: - 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. - 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 $P=NP$. Chen et al. [SIDMA, 2009] %[On the Approximability of Influence in Social Networks. SIAM Journal on Discrete Mathematics, 23(3):1400-1415, 2009] had shown that the TSS Problem on general graphs does not admit an approximation algorithm with a performance guarantee asymptotically better than $O(2^{\log^{1 - \epsilon} n})$, where $n$ is the number of vertices of the graph under consideration, unless $NP \subseteq DTIME(n^{polylog(n)})$.Comment: 11 page