An Approximation Algorithm for the Vertex Multicut on Trees with an Application to the Tracking Paths Problem

This paper considers two problems related to the selection of weighted vertices to cover a set of paths. The first problem is the Vertex Multicut on Trees whose goal is to find the cheapest set of vertices that cut every given pairs of vertices. Another problem is the Tracking Paths problem where one would like to choose a set of “beacons” in the network so that every distinct path from source $s$ to target $t$ can be uniquely identified by an intersection pattern with these beacons. Formally, we present a 2-approximation algorithm for the Weighted Vertex Multicut on Trees based on a standard randomized rounding procedure. This algorithm can be used as a subroutine in a recent approximation algorithm for the Tracking Paths problem by Blaže], Choudhary, Knop, Křišt'an, Suchý, and Valla [WAOA'21], improving the approximation ratio from 66 to 6. We note that Blažej et. al. also independently obtained a similar improvement.