Improved kernels for tracking paths

Tracking of moving objects is crucial to security systems and networks. Given a graph G, terminal vertices s and t, and an integer k, the TRACKING PATHS problem asks whether there exists at most k vertices, which if marked as trackers, would ensure that the sequence of trackers encountered in each s-t path is unique. It is known that the problem is NP-hard and admits a kernel (reducible to an equivalent instance) with O(k⁶) vertices and O(k⁷) edges, when parameterized by the size of the output (tracking set) k [4]. In this paper we improve the size of the kernel substantially by providing a kernel with O(k²) vertices and edges for general graphs and a kernel with O(k) vertices and edges for planar graphs. We do this via a new concept, namely a tree-sink structure. We also show that finding a tracking set of size at most n−k for a graph on n vertices is hard for the parameterized complexity class W[1], when parameterized by k.