Constant Factor Approximation for Tracking Paths and 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 66-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 2 ) approximation algorithm for r-Fault Tolerant Feedback Vertex Set if r is a constant.⋆ The authors acknowledge the support of the OP VVV MEYS funded project CZ.