Centroidal localization game

One important problem in a network is to locate an (invisible) moving entity by using distance-detectors placed at strategical locations. For instance, the metric dimension of a graph $G$ is the minimum number $k$ of detectors placed in some vertices $\{v_1,\cdots,v_k\}$ such that the vector $(d_1,\cdots,d_k)$ of the distances $d(v_i,r)$ between the detectors and the entity's location $r$ allows to uniquely determine $r \in V(G)$. In a more realistic setting, instead of getting the exact distance information, given devices placed in $\{v_1,\cdots,v_k\}$, we get only relative distances between the entity's location $r$ and the devices (for every $1\leq i,j\leq k$, it is provided whether $d(v_i,r) >$, $<$, or $=$ to $d(v_j,r)$). The centroidal dimension of a graph $G$ is the minimum number of devices required to locate the entity in this setting. We consider the natural generalization of the latter problem, where vertices may be probed sequentially until the moving entity is located. At every turn, a set $\{v_1,\cdots,v_k\}$ of vertices is probed and then the relative distances between the vertices $v_i$ and the current location $r$ of the entity are given. If not located, the moving entity may move along one edge. Let $\zeta^* (G)$ be the minimum $k$ such that the entity is eventually located, whatever it does, in the graph $G$. We prove that $\zeta^* (T)\leq 2$ for every tree $T$ and give an upper bound on $\zeta^*(G\square H)$ in cartesian product of graphs $G$ and $H$. Our main result is that $\zeta^* (G)\leq 3$ for any outerplanar graph $G$. We then prove that $\zeta^* (G)$ is bounded by the pathwidth of $G$ plus 1 and that the optimization problem of determining $\zeta^* (G)$ is NP-hard in general graphs. Finally, we show that approximating (up to any constant distance) the entity's location in the Euclidean plane requires at most two vertices per turn

Sequential Metric Dimension

International audienceSeager 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 $k ≥ 1$ 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 by probing at most $k$ vertices per step. We show this problem is NP-complete when $k$ (resp.,) is a fixed parameter. Our main results are for the class of trees where we prove this problem is NP-complete when $k$ and 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

Sequential Metric Dimension

International audienceIn 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 $k\geq 1$ vertices can be probed at every step. Our game also generalizes the notion of the {\it metric dimension} of a graph.Precisely, given a graph $G$ and two integers $k,\ell \geq 1$, the {\sc Localization} problem asks whether there exists a strategy to locate a target hidden in $G$ in at most $\ell$ steps and probing at most $k$ vertices per step. We first show that, in general, this problem is \textsf{NP}-complete for every fixed $k \geq 1$ (resp., $\ell \geq 1$).We then focus on the class of trees.On the negative side,we prove that the \Localization problem is \textsf{NP}-complete in trees when $k$ and $\ell$ are part of the input. On the positive side, we design a $(+1)$-approximation algorithm for the problem in $n$-node trees, {\it 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.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 {\it centroidal dimension} of a graph

Cooperative Estimation via Altruism

A novel approach, based on the notion of altruism, is presented to cooperative estimation in a system comprising two information-sharing estimators. The underlying assumption is that the system's global mission can be accomplished even if only one of the estimators achieves satisfactory performance. The notion of altruism motivates a new definition of cooperative estimation optimality that generalizes the common definition of minimum mean square error optimality. Fundamental equations are derived for two types of altruistic cooperative estimation problems, corresponding to heterarchical and hierarchical setups. Although these equations are hard to solve in the general case, their solution in the Gaussian case is straightforward and only entails the largest eigenvalue of the conditional covariance matrix and its corresponding eigenvector. Moreover, in that case the performance improvement of the two altruistic cooperative estimation techniques over the conventional (egoistic) estimation approach is shown to depend on the problem's dimensionality and statistical distribution. In particular, the performance improvement grows with the dispersion of the spectrum of the conditional covariance matrix, rendering the new estimation approach especially appealing in ill-conditioned problems. The performance of the new approach is demonstrated using a numerical simulation study.Comment: 14 pages, 9 figure

Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters

For a graph $G$, a subset $S \subseteq V(G)$ is called a \emph{resolving set}if for any two vertices $u,v \in V(G)$, there exists a vertex $w \in S$ suchthat $d(w,u) \neq d(w,v)$. The {\sc Metric Dimension} problem takes as input agraph $G$ and a positive integer $k$, and asks whether there exists a resolvingset of size at most $k$. This problem was introduced in the 1970s and is knownto be NP-hard~[GT~61 in Garey and Johnson's book]. In the realm ofparameterized complexity, Hartung and Nichterlein~[CCC~2013] proved that theproblem is W[2]-hard when parameterized by the natural parameter $k$. They alsoobserved that it is FPT when parameterized by the vertex cover number and askedabout its complexity under \emph{smaller} parameters, in particular thefeedback vertex set number. We answer this question by proving that {\sc MetricDimension} is W[1]-hard when parameterized by the feedback vertex set number.This also improves the result of Bonnet and Purohit~[IPEC 2019] which statesthat the problem is W[1]-hard parameterized by the treewidth. Regarding theparameterization by the vertex cover number, we prove that {\sc MetricDimension} does not admit a polynomial kernel under this parameterizationunless $NP\subseteq coNP/poly$. We observe that a similar result holds when theparameter is the distance to clique. On the positive side, we show that {\scMetric Dimension} is FPT when parameterized by either the distance to clusteror the distance to co-cluster, both of which are smaller parameters than thevertex cover number.<br