TY - JOUR

T1 - Tracking Paths

AU - Banik, Aritra

AU - Katz, Matthew J.

AU - Packer, Eli

AU - Simakov, Marina

N1 - Funding Information:
The authors would like to thank the anonymous reviewers whose comments have greatly improved this manuscript. Work on this paper by M. Katz was supported by grant 1884/16 from the Israel Science Foundation .
Publisher Copyright:
© 2019 Elsevier B.V.

PY - 2020/8/15

Y1 - 2020/8/15

N2 - 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.

AB - 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.

KW - Guarding

KW - Shortest paths

KW - Target tracking

KW - Tracking paths

UR - http://www.scopus.com/inward/record.url?scp=85076206902&partnerID=8YFLogxK

U2 - 10.1016/j.dam.2019.11.013

DO - 10.1016/j.dam.2019.11.013

M3 - Article

AN - SCOPUS:85076206902

VL - 282

SP - 22

EP - 34

JO - Discrete Applied Mathematics

JF - Discrete Applied Mathematics

SN - 0166-218X

ER -