The minimum tracking set problem is an optimization problem that deals with monitoring communication paths that can be used for exchanging point-to-point messages using as few tracking devices as possible. More precisely, a tracking set of a given graph G and a set of source-destination pairs of vertices, is a subset T of vertices of G such that the vertices in T traversed by any source-destination shortest path P uniquely identify P. The minimum tracking set problem has been introduced in [Banik et al., CIAC 2017] for the case of a single source-destination pair. There, the authors show that the problem is APX-hard and that it can be 2-approximated for the class of planar graphs, even though no hardness result is known for this case. In this paper we focus on the case of multiple source-destination pairs and we present the first \(\widetilde{O}(\sqrt{n})\)-approximation algorithm for general graphs. Moreover, we prove that the problem remains NP-hard even for cubic planar graphs and all pairs \(S \times D\), where S and D are the sets of sources and destinations, respectively. Finally, for the case of a single source-destination pair, we design an (exact) FPT algorithm w.r.t. the maximum number of vertices at the same distance from the source.
[1]
Guido Proietti,et al.
Network verification via routing table queries
,
2015,
J. Comput. Syst. Sci..
[2]
Thomas Erlebach,et al.
Network Discovery and Verification
,
2005,
IEEE Journal on Selected Areas in Communications.
[3]
Vijay V. Vazirani,et al.
Approximation Algorithms
,
2001,
Springer Berlin Heidelberg.
[4]
Azriel Rosenfeld,et al.
Landmarks in Graphs
,
1996,
Discret. Appl. Math..
[5]
David Lichtenstein,et al.
Planar Formulae and Their Uses
,
1982,
SIAM J. Comput..
[6]
Saket Saurabh,et al.
A Polynomial Sized Kernel for Tracking Paths Problem
,
2018,
LATIN.
[7]
David Eppstein,et al.
Tracking Paths in Planar Graphs
,
2019,
ISAAC.
[8]
Aritra Banik,et al.
Fixed-Parameter Tractable Algorithms for Tracking Set Problems
,
2018,
CALDAM.