In $\textit{total domination}$, given a graph $G=(V,E)$, we seek a minimum-size set of nodes $S\subseteq V$, such that every node in $V$ has at least one neighbor in $S$. We define a $\textit{fault-tolerant}$ version of total domination, where we require any node in $V \setminus S$ to have at least $m$ neighbors in $S$. Let $Δ$ denote the maximum degree in $G$. We prove a first $1 + \ln(Δ+ m - 1)$ approximation for fault-tolerant total domination. We also consider fault-tolerant variants of the weighted $\textit{partial positive influence dominating set}$ problem, where we seek a minimum-size set of nodes $S\subseteq V$, such that every node in $V$ is either a member of $S$ or the sum of weights of its incident edges leading to nodes in $S$ is at least half of the sum of weights over all its incident edges. We prove the first logarithmic approximations for the simple, total, and connected variants of this problem. To prove the result for the connected case, we extend the general approximation framework for non-submodular functions from integer-valued to fractional-valued functions, which we believe is of independent interest.

翻译：在$\textit{全支配}$问题中，给定图$G=(V,E)$，我们寻求一个最小规模的节点集合$S\subseteq V$，使得$V$中的每个节点在$S$中至少有一个邻居。我们定义了一个$\textit{容错}$版本的全支配问题，要求$V \setminus S$中的任意节点在$S$中至少有$m$个邻居。令$Δ$表示$G$中的最大度数。我们证明了容错全支配问题的首个$1 + \ln(Δ+ m - 1)$近似比。我们还考虑了加权$\textit{部分正影响支配集}$问题的容错变体，该问题要求寻找一个最小规模的节点集合$S\subseteq V$，使得$V$中的每个节点要么属于$S$，要么其连接到$S$中节点的关联边权重之和至少为其所有关联边总权重的一半。我们针对该问题的简单、全连通及连通变体给出了首个对数近似算法。为证明连通情况下的结果，我们将非子模函数的通用近似框架从整数值函数推广至分数值函数，我们认为这一推广具有独立的研究价值。