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(r) approximation algorithm for r-Fault Tolerant Feedback Vertex Set if r is a constant.
@article{arxiv.2108.01430,
title = {Constant Factor Approximation for Tracking Paths and Fault Tolerant Feedback Vertex Set},
author = {Václav Blažej and Pratibha Choudhary and Dušan Knop and Jan Matyáš Křišťan and Ondřej Suchý and Tomáš Valla},
journal= {arXiv preprint arXiv:2108.01430},
year = {2022}
}