Polynomial Time Algorithms for Tracking Path Problems

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 Δ ≥ 6 , and give a 2 ( Δ + 1 ) -approximate algorithm for this case. Further, we give a polynomial time algorithm which, given an undirected graph G , a tracking set T ⊆ V ( G ) , and a sequence of trackers π , returns the unique s - t path in G that corresponds to π , 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.