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We consider the parameterized complexity of the problem of tracking shortest s-t paths in graphs, motivated by applications in security and wireless networks.Given an undirected and unweighted graph with a source s and a destination t, Tracking Shortest Paths asks if there exists a k-sized subset of vertices (referred to as tracking set ) that intersects each shortest s-t path in a distinct set of vertices.We first generalize this problem for set systems, namely Tracking Set System, where given a family of subsets of a universe, we are required to find a subset of elements from the universe that has a unique intersection with each set in the family. Tracking Set System is shown to be fixed-parameter tractable due to its relation with a known problem, Test Cover. By a reduction to the well-studied d-hitting set problem, we give a polynomial (with respect to k) ✩ Under review in a journal. A preliminary version of the paper appeared in the proceedings of CALDAM 2018 [1]

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 $$s$$ s -$$t$$ 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 $$\Delta \ge 6$$ Δ ≥ 6 , and give a $$2(\Delta +1)$$ 2 ( Δ + 1 ) -approximate algorithm for this case. Further, we give a polynomial time algorithm which, given an undirected graph G, a tracking set $$T\subseteq V(G)$$ T ⊆ V ( G ) , and a sequence of trackers $$\pi $$ π , returns the unique $$s$$ s -$$t$$ t path in G that corresponds to $$\pi $$ π , if one exists. Finally we analyze the version of tracking $$s$$ s -$$t$$ t paths where paths are tracked using edges instead of vertices, and we give a polynomial time algorithm for the same.

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