Abstract
We consider several problems dealing with tracking of moving objects (e.g., vehicles) in networks. Given a graph \(G=(V,E)\) and two vertices \(s,t \in V\), a set of vertices \(T \subseteq V\) is a tracking set for G (w.r.t. paths from s to t), if one can distinguish between any two paths from s to t by the order in which the vertices of T appear (or do not appear) in them. We prove that the problem of finding a minimum-cardinality tracking set w.r.t. shortest paths from s to t is NP-hard and even APX-hard. On the other hand, for the common case where G is planar, we present a 2-approximation algorithm for this problem. We also consider the following related problem: Given a graph G, two vertices s and t, and a set of forbidden vertices \(V_F \subseteq V-\{s,t\}\), find a minimum-cardinality set of trackers \(V^* \subset V\), such that a shortest path P from s to t passes through a forbidden vertex if and only if it passes through a vertex of \(V^*\). We present a polynomial-time (exact) algorithm for this problem.
M.J. Katz—Supported by grant 1884/16 from the Israel Science Foundation.
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Banik, A., Katz, M.J., Packer, E., Simakov, M. (2017). Tracking Paths. In: Fotakis, D., Pagourtzis, A., Paschos, V. (eds) Algorithms and Complexity. CIAC 2017. Lecture Notes in Computer Science(), vol 10236. Springer, Cham. https://doi.org/10.1007/978-3-319-57586-5_7
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DOI: https://doi.org/10.1007/978-3-319-57586-5_7
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