Polynomial Time Algorithms for Tracking Path Problems

Abstract

Given a graph G, and terminal vertices s and t, the Tracking Paths problem asks to compute a set of minimum number of vertices to be marked as trackers, such that the sequence of trackers encountered in each s-t path is unique. Tracking Paths is NP-hard in both directed and undirected graphs in general. In this paper we give a collection of polynomial time algorithms for some restricted versions of Tracking Paths. We prove that Tracking Paths is polynomial time solvable for undirected chordal graphs and tournament graphs. We also show that Tracking Paths is NP-hard in graphs with bounded maximum degree Delta ge 6, and give a 2(Delta +1)-approximate algorithm for this case. Further, we give a polynomial time algorithm which, given an undirected graph G, a tracking set Tsubseteq V(G), and a sequence of trackers pi , returns the unique s-t path in G that corresponds to pi , if one exists. Finally we analyze the version of tracking s-t paths where paths are tracked using edges instead of vertices, and we give a polynomial time algorithm for the same.