Abstract
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.
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Notes
- 1.
Observe that a TS always exists unless \(\mathcal {P}\) contains two pairs of the form (s, t) and (t, s). We then assume that our TS instances never contain such pairs.
- 2.
Observe that it is not possible to reduce the multi-source multi-destination case to the single-pair case by simply adding a super-source and a super-destination connected to all the sources and all the destinations, respectively, as erroneously claimed in [4].
- 3.
The \(O^*\) notation suppresses polynomial multiplicative factors w.r.t. n.
- 4.
Notice that, as a consequence of this definition, the pair \((P_1, P_2)\) in which \(P_1\) and \(P_2\) coincide and consist of the single edge (u, v) is independent.
- 5.
When the graph G is clear from context, we may omit the subscript.
- 6.
This means that there may be two paths, one in \(P(s_1,t)\) and the other one in \(P(s_2,t)\) that traverse the same subset of vertices of the tracking set.
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Bilò, D., Gualà, L., Leucci, S., Proietti, G. (2019). Tracking Routes in Communication Networks. In: Censor-Hillel, K., Flammini, M. (eds) Structural Information and Communication Complexity. SIROCCO 2019. Lecture Notes in Computer Science(), vol 11639. Springer, Cham. https://doi.org/10.1007/978-3-030-24922-9_6
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