Constant factor approximation for tracking paths and fault tolerant feedback vertex set

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 $O\left(r\right)$ approximation algorithm for r-Fault Tolerant Feedback Vertex Set if $r$ is a constant.