A widely studied model for influence diffusion in socialnetworks are target sets. For a graph $G$ and an integer-valued threshold function $\tau$ on its vertex set, a target set or dynamic monopoly is a set of vertices of $G$ such that iteratively adding to it vertices $u$ of $G$ that have at least $\tau \left(u\right)$ neighbors in it eventually yields the entire vertex set of $G$. This notion is limited to the binary choice of including a vertex in the target set or not, and Cordasco, Gargano, Rescigno, and Vaccaro proposed partial incentives as a variant allowing for intermediate choices.

We show that finding optimal partial incentives is hard for chordal graphs and planar graphs but tractable for graphs of bounded treewidth and for interval graphs with bounded thresholds. We also contribute some new results about target set selection on planar graphs by showing the hardness of this problem, and by describing an efficient $\mathcal{O}\left(\sqrt{n}\right)$-approximation algorithm as well as a PTAS for the dual problem of finding a maximum degenerate set.

广泛研究的社会网络影响力扩散模型是目标集。对于图$G$ 和一个整数阈值函数 $\tau$在其顶点集上，目标集或动态垄断是指一组顶点$G$ 这样反复添加顶点 $ü$ 的 $G$ 至少有 $\tau （ü）$ 最终它的邻居产生了整个顶点集 $G$。这个概念仅限于在目标集中是否包含一个顶点的二元选择，Cordasco，Gargano，Rescigno和Vaccaro提出了部分激励作为允许中间选择的变体。

我们表明，对于弦图和平面图很难找到最优的部分激励，但是对于有界树宽图和具有有界阈值的区间图则很难处理。通过显示此问题的难度并描述有效的方法，我们还为平面图上的目标集选择提供了一些新结果。$\mathcal{Ø}（\sqrt{ñ}）$-近似算法以及PTAS用于发现最大简并集的双重问题。