Tracking Paths

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∈V, 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 describe how to preprocess G (and its tracking set w.r.t. shortest paths from s to t), so that given a sequence of visited trackers, one can reconstruct the traversed path P in O(|P|) time. Moreover, we consider the following related problem: Given a graph G, two vertices s and t, and a set of forbidden vertices F⊆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.