Constant Factor Approximation for Tracking Paths and Fault Tolerant Feedback Vertex Set

TL;DR

This paper presents the first constant factor approximation algorithms for the Tracking Paths problem in weighted and unweighted graphs, and for the r-Fault Tolerant Feedback Vertex Set problem, advancing solutions for these complex graph problems.

Contribution

It introduces the first constant factor approximation algorithms for Tracking Paths and provides an al(r) approximation for r-Fault Tolerant Feedback Vertex Set with fixed r.

Findings

6-approximation for weighted Tracking Paths

4-approximation for unweighted Tracking Paths

al(r) approximation for r-Fault Tolerant Feedback Vertex Set

Abstract

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.