# 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 > 1/2d_v for every v ∈ V. In this special case, TSS can be approximated within a ratio of O(Δ), where Δ = max_v ∈ 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∈ V t_v > 1/2∑_v ∈ V d_v. We design approximation algorithms for some subclasses of CoA. For example, if t_v ≥1/2d_v for every v ∈ V, we can find in polynomial time a contagious set of size Õ(Δ· OPT^2 ), 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, and approximation algorithms for a related problem, that of TSS with partial incentives.

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