Sequential Metric Dimension

Abstract

Seager introduced the following game in 2013. An invisible and immobile target is hidden at some vertex of a graph G. Every step, one vertex v of G can be probed which results in the knowledge of the distance between v and the target. The objective of the game is to minimize the number of steps needed to locate the target, wherever it is.

We address the generalization of this game where $k\ge 1$ 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 $k,\ell \ge 1$, the Localization Problem asks whether there exists a strategy to locate a target hidden in G in at most $\ell$ steps by probing at most k vertices per step. We show this problem is NP-complete when k (resp., $\ell$) is a fixed parameter.

Our main results are for the class of trees where we prove this problem is NP-complete when k and $\ell$ are part of the input but, despite this, we design a polynomial-time $(+1)$-approximation algorithm in trees which gives a solution using at most one more step than the optimal one. It follows that the Localization Problem is polynomial-time solvable in trees if k is fixed.

This work has been partially supported by ANR program “Investments for the Future” under reference ANR-11-LABX-0031-01, the Inria Associated Team AlDyNet. Due to lack of space, several proofs have been omitted and can be found in [3].