Solving Target Set Selection with Bounded Thresholds Faster than $$2^n$$

In this paper we consider the Target Set Selection problem. The problem naturally arises in many fields like economy, sociology, medicine. one is given a graph G with function $${{\,\mathrm{thr}\,}}: V(G) \rightarrow {\mathbb {N}} \cup \{0\}$$ and two integers $$k, \ell $$ . goal of to activate at most k vertices initially so that end activation process there are least $$\ell activated vertices. occurs following way: (i) once activated, vertex stays forever; (ii) v becomes if $${{\,\mathrm{thr}\,}}(v)$$ its neighbours activated. different special cases were extensively studied from approximation parameterized points view. For example, parameterizations by parameters studied: treewidth, feedback set, diameter, size target cover, cluster editing number others. Despite extensive study it still unknown whether can be solved $${\mathcal {O}}^*\left( (2-\epsilon )^n\right) time for some $$\epsilon >0$$ We partially answer question presenting several faster-than-trivial algorithms work constant thresholds, dual thresholds or when threshold value each bounded one-third degree. Also, show W[1]-hard even all constant.