Sequential Metric Dimension

Julien Bensmail 1 Dorian Mazauric 2 Fionn Mc Inerney 1 Nicolas Nisse 1 Stéphane Perennes 1
1 COATI - Combinatorics, Optimization and Algorithms for Telecommunications
CRISAM - Inria Sophia Antipolis - Méditerranée , COMRED - COMmunications, Réseaux, systèmes Embarqués et Distribués
2 ABS - Algorithms, Biology, Structure
CRISAM - Inria Sophia Antipolis - Méditerranée
Abstract : In the localization game, introduced by Seager in 2013, an invisible and immobile target is hidden at some vertex of a graph G. At every step, one vertex v of G can be probed which results in the knowledge of the distance between v and the secret location of the target. The objective of the game is to minimize the number of steps needed to locate the target whatever be its location. We address the generalization of this game where k ≥ 1 vertices can be probed at every step. Our game also generalizes the notion of the metric dimension of a graph. Precisely, given a graph G and two integers k, ≥ 1, the Localization Problem asks whether there exists a strategy to locate a target hidden in G in at most steps and probing at most k vertices per step. We first show that this problem is NP-complete when k (resp.,) is a fixed parameter. Our main results concern the study of the Localization Problem in the class of trees. We prove that this problem is NP-complete in trees when k and are part of the input. On the positive side, we design a (+1)-approximation in n-node trees, i.e., an algorithm that computes in time O(n log n) (independent of k) a strategy to locate the target in at most one more step than an optimal strategy. This algorithm can be used to solve the Localization Problem in trees in polynomial-time if k is fixed.
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[Research Report] Inria. 2018
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Julien Bensmail, Dorian Mazauric, Fionn Mc Inerney, Nicolas Nisse, Stéphane Perennes. Sequential Metric Dimension. [Research Report] Inria. 2018. 〈hal-01717629〉

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