Tracking paths

We consider several problems dealing with tracking of mov-ing objects (e.g., vehicles) in networks. Given a graph G = (V, E) and two vertices (formula presented), a set of vertices T ⊆ 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 VF⊆ V − {s, t}, find a minimum-cardinality set of trackers V∗⊂ 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.