Target set selection for conservative populations

Let G=(V,E) be a graph on n vertices, where d(v) denotes the degree of vertex v, and t(v) is a threshold associated with v. We consider a process in which initially a set S of vertices becomes active, and thereafter, in discrete time steps, every vertex v that has at least t(v) active neighbors becomes active as well. The set S is contagious if eventually all V becomes active. The target set selection problem TSS asks for the smallest contagious set. TSS is NP-hard and moreover, notoriously difficult to approximate. In the conservative special case of TSS, t(v)>12d(v) for every v∈V. In this special case, TSS can be approximated within a ratio of O(Δ), where Δ=maxv∈V[d(v)]. In this work we introduce a more general class of TSS instances that we refer to as conservative on average (CoA), that satisfy the condition ∑v∈Vt(v)>12∑v∈Vd(v). We design approximation algorithms for some subclasses of CoA. For example, if t(v)≥12d(v) for every v∈V, we can find in polynomial time a contagious set of size ÕΔ⋅OPT2, where OPT is the size of a smallest contagious set in G. We also provide several hardness of approximation results. For example, assuming the unique games conjecture, we prove that TSS on CoA instances with Δ≤3 cannot be approximated within any constant factor. We also present results concerning the fixed parameter tractability of CoA TSS instances.