跟踪路径的常数因子逼近与容错反馈顶点集
Constant Factor Approximation for Tracking Paths and Fault Tolerant Feedback Vertex Set
摘要：
考虑具有源$s$和目标$t$的顶点加权图$G$。跟踪路径需要找到顶点（跟踪器）的最小权重集合，使得从$s$到$t$的每条路径中的跟踪器序列是唯一的。在这项工作中，我们推导出一个因子$6$ -近似算法跟踪加权图中的路径和一个因子$4$ -近似算法，如果输入是不加权的。这是这个问题的第一个常数因子近似。同时，我们还研究了与之密切相关的$r$ -容错反馈顶点集问题的近似。在那里，对于一个固定的整数$r$和一个给定的顶点加权图$G$，任务是找到一个最小的权重集的顶点相交的$G$的每一个周期至少$r+1$顶点。当$r$为常数时，给出了$r$ -容错反馈顶点集的因子$\mathcal{O}(r)$近似算法.
英文摘要：
Consider a vertex-weighted graph $G$ with a source $s$ and a target $t$. Tracking Paths requires finding a minimum weight set of vertices (trackers) such that the sequence of trackers in each path from $s$ to $t$ is unique. In this work, we derive a factor $6$-approximation algorithm for Tracking Paths in weighted graphs and a factor $4$-approximation algorithm if the input is unweighted. This is the first constant factor approximation for this problem. While doing so, we also study approximation of the closely related $r$-Fault Tolerant Feedback Vertex Set problem. There, for a fixed integer $r$ and a given vertex-weighted graph $G$, the task is to find a minimum weight set of vertices intersecting every cycle of $G$ in at least $r+1$ vertices. We give a factor $\mathcal{O}(r)$ approximation algorithm for $r$-Fault Tolerant Feedback Vertex Set if $r$ is a constant.
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