Sequential Metric Dimension

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 rst show that, in general, this problem is NP-complete for every xed k ≥ 1 (resp., ≥ 1). We then focus on the class of trees. On the negative side, we prove that the Localization problem is NPcomplete in trees when k and are part of the input. On the positive side, we design a (+1)-approximation algorithm for the problem 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 xed. We also consider some of these questions in the context where, upon probing the vertices, the relative distances to the target are retrieved. This variant of the problem generalizes the notion of the centroidal dimension of a graph.

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“…4. The root r of T has eight children (3,3), (3, 3), (2, 2), (2, 2), (1, 1), and (0, 0), respectively. Let k = 4.…”

Section: A Polynomial-time Algorithm For the Next Stepsmentioning

confidence: 99%

Sequential Metric Dimension

et al. 2020

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“…In the combinatorics literature, this adaptive version of the MD was introduced in [22] under the name sequential location number. The same notion was later referred to as the sequential metric dimension (SMD) in [3]. We focus on the Erdős–Rényi random graph model, because of the previous results on the MD of Erdős–Rényi graphs of Bollobás, Mitsche, and Prałat [4].…”

Section: Introductionmentioning

confidence: 99%

Sequential metric dimension for random graphs

Ódor

Thiran

2021

J. Appl. Probab.

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In the localization game on a graph, the goal is to find a fixed but unknown target node $v^\star$ with the least number of distance queries possible. In the jth step of the game, the player queries a single node $v_j$ and receives, as an answer to their query, the distance between the nodes $v_j$ and $v^\star$ . The sequential metric dimension (SMD) is the minimal number of queries that the player needs to guess the target with absolute certainty, no matter where the target is.The term SMD originates from the related notion of metric dimension (MD), which can be defined the same way as the SMD except that the player’s queries are non-adaptive. In this work we extend the results of Bollobás, Mitsche, and Prałat [4] on the MD of Erdős–Rényi graphs to the SMD. We find that, in connected Erdős–Rényi graphs, the MD and the SMD are a constant factor apart. For the lower bound we present a clean analysis by combining tools developed for the MD and a novel coupling argument. For the upper bound we show that a strategy that greedily minimizes the number of candidate targets in each step uses asymptotically optimal queries in Erdős–Rényi graphs. Connections with source localization, binary search on graphs, and the birthday problem are discussed.

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Sequential Metric Dimension

Bensmail

Mazauric

Inerney

et al. 2018

Approximation and Online Algorithms

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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 rst show that, in general, this problem is NP-complete for every xed k ≥ 1 (resp., ≥ 1). We then focus on the class of trees. On the negative side, we prove that the Localization problem is NPcomplete in trees when k and are part of the input. On the positive side, we design a (+1)-approximation algorithm for the problem 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 xed.We also consider some of these questions in the context where, upon probing the vertices, the relative distances to the target are retrieved. This variant of the problem generalizes the notion of the centroidal dimension of a graph.

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Sequential Metric Dimension

Cited by 3 publications

References 17 publications

Sequential Metric Dimension

Sequential Metric Dimension

Sequential metric dimension for random graphs

Sequential Metric Dimension

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