Section: New Results

Games on Graphs

Participants : Julien Bensmail, Nicolas Nisse, Fionn Mc Inerney, Stéphane Pérennes.

We study several two-player games on graphs. Some of these games allow to model real-life applications. In the case of the Spy-game presented below, we propose a successful new approach by studying fractional relaxation of such games.

Spy-game on graphs and eternal domination

In [24] we define and study the following two-player game on a graph $G$. Let $k\in {𝐍}^{*}$. A set of $k$ guards is occupying some vertices of $G$ while one spy is standing at some node. At each turn, first the spy may move along at most s edges, where $s\in {𝐍}^{*}$ is his speed. Then, each guard may move along one edge. The spy and the guards may occupy the same vertices. The spy has to escape the surveillance of the guards, i.e., must reach a vertex at distance more than $d\in 𝐍$ (a predefined distance) from every guard. Can the spy win against $k$ guards? Similarly, what is the minimum distance $d$ such that $k$ guards may ensure that at least one of them remains at distance at most $d$ from the spy? This game generalizes two well-studied games: Cops and robber games (when $s=1$) and Eternal Dominating Set (when $s$ is unbounded).

In [24], we consider the computational complexity of the problem, showing that it is NP-hard (for every speed s and distance d) and that some variant of it is PSPACE-hard in DAGs. Then, we establish tight tradeoffs between the number of guards, the speed s of the spy and the required distance d when $G$ is a path or a cycle.

In order to determine the smallest number of guards necessary for this task, we analyze in [25] the game through a Linear Programming formulation and the fractional strategies it yields for the guards. We then show the equivalence of fractional and integral strategies in trees. This allows us to design a polynomial-time algorithm for computing an optimal strategy in this class of graphs. Using duality in Linear Programming, we also provide non-trivial bounds on the fractional guard-number of grids and torus. We believe that the approach using fractional relaxation and Linear Programming is promising to obtain new results in the field of combinatorial games.

In [60] we pursue the study of the eternal domination game (which is equivalent to the spy game when $s$ is unbounded and $d=0$) on strong grids $P_n\square P_m$. Cartesian grids $P_n\square P_m$ have been vastly studied with tight bounds existing for small grids such as $k\times n$ grids for $k\in \{2,3,4,5\}$. It was recently proven that ${\gamma }_{all}^{\infty }(P_n\square P_m)=\gamma (P_n\square P_m)+O(n+m)$ where $\gamma (P_n\square P_m)$ is the domination number of $P_n\square P_m$ which lower bounds the eternal domination number. We prove that, for all $n,m\in {\mathbb{N}}^{*}$ such that $m\ge n$, $\lceil \frac{nm}{9}\rceil +\Omega (n+m)={\gamma }_{all}^{\infty }(P_n⊠P_m)=\lceil \frac{nm}{9}\rceil +O\left(m\sqrt{n}\right)$ (note that $\lceil \frac{nm}{9}\rceil$ is the domination number of $P_n⊠P_m$).

Metric dimension & localization

The questions that we study there are variant of the usual Metric Dimension problem in which one wishes to identify the vertices of a graph from the knowledge of the distances to a few points. This is motivated by localization problems, e.g., in cellular networks. few anchors.

In [19] we introduce a generalization of metric dimension based on a pursuit graph game that resembles the famous Cops and Robbers game. In this game, an invisible target is hidden at some vertex of a graph (at each turn, it may move to a neighbor). At every step, $k\ge 1$ vertices of $G$ can be probed which results in the knowledge of the distances between each of these vertices and the secret location of the target. We provide upper bounds on the related graph invariant $\zeta \left(G\right)$, defined as the least number of probes per turn needed to localize the robber on a graph $G$, for several classes of graphs (trees, bipartite graphs, etc). Our main result is that, surprisingly, there exists planar graphs of treewidth 2 and unbounded $\zeta \left(G\right)$. On a positive side, we prove that $\zeta \left(G\right)$ is bounded by the pathwidth of $G$. We then show that the algorithmic problem of determining $\zeta \left(G\right)$ is NP-hard in graphs with diameter at most 2. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane. We further study this problem in [18] where, in particular, we prove that $\zeta \left(G\right)\le 3$ in outer-planar graphs.

In [39], [56], [38], we address the sequential metric dimension when the invisible target is immobile. The objective of the game is to minimize the number of steps needed to locate the target whatever be its location. 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 and probing at most $k$ vertices per step. We first show that, in general, this problem is NP-complete for every fixed $k\ge 1$ (resp., $\ell \ge 1$). We then focus on the class of trees. On the negative side, we prove that the Localization problem is NP-complete in trees when $k$ and $\ell$ are part of the input. On the positive side, we design a $(+1)$-approximation for the problem in $n$-node trees, i.e., an algorithm that computes in time $O(nlogn)$ (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.

In [57] we try to understand the phenomena when one choose an orientation of an (undirected) graphs. Namely, we study, for particular graph families, the maximum metric dimension over all strongly-connected orientations, by exhibiting lower and upper bounds on this value. We first exhibit general bounds for graphs with bounded maximum degree. In particular, we prove that, in the case of subcubic $n$-node graphs, all strongly-connected orientations asymptotically have metric dimension at most $\frac{n}{2}$, and that there are such orientations having metric dimension $\frac{2n}{5}$. We then consider strongly-connected orientations of grids. For a torus with $n$ rows and $m$ columns, we show that the maximum value of the metric dimension of a strongly-connected Eulerian orientation is asymptotically $\frac{nm}{2}$ (the equality holding when $n,m$ are even, which is best possible). For a grid with $n$ rows and $m$ columns, we prove that all strongly-connected orientations asymptotically have metric dimension at most $\frac{2nm}{3}$, and that there are such orientations having metric dimension $\frac{nm}{2}$.

Orienting edges to fight fire in graphs

In [12], we investigate a new oriented variant of the Firefighter Problem. In the traditional Firefighter Problem, a fire breaks out at a given vertex of a graph, and at each time interval spreads to neighbouring vertices that have not been protected, while a constant number of vertices are protected at each time interval. In our version of the problem, the firefighters are able to orient the edges of the graph before the fire breaks out, but the fire could start at any vertex. We consider this problem when played on a graph in one of several graph classes, and give upper and lower bounds on the number of vertices that can be saved. In particular, when one firefighter is available at each time interval, and the given graph is a complete graph, or a complete bipartite graph, we present firefighting strategies that are provably optimal. We also provide lower bounds on the number of vertices that can be saved as a function of the chromatic number, of the maximum degree, and of the treewidth of a graph. For a sub-cubic graph, we show that the firefighters can save all but two vertices, and this is best possible.

Network decontamination

The Network Decontamination problem consists in coordinating a team of mobile agents in order to clean a contaminated network. The problem is actually equivalent to tracking and capturing an invisible and arbitrarily fast fugitive. This problem has natural applications in network security in computer science or in robotics for search or pursuit-evasion missions. In this Chapter, we focus on networks modeled by graphs. Many different objectives have been studied in this context, the main one being the minimization of the number of mobile agents necessary to clean a contaminated network. Another important aspect is that this optimization problem has a deep graph-theoretical interpretation. Network decontamination and, more precisely, graph searching models, provide nice algorithmic interpretations of fundamental concepts in the Graph Minors theory by Robertson and Seymour. For all these reasons, graph searching variants have been widely studied since their introduction by Breish (1967) and mathematical formalizations by Parsons (1978) and Petrov (1982). Our chapter [61] consists of an overview of algorithmic results on graph decontamination and graph searching.

Hyperopic Cops and Robbers

We introduce in [17] a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop. The hyperopic cop number is the corresponding analogue of the cop number, and we investigate bounds and other properties of this parameter. We characterize the cop-win graphs for this variant, along with graphs with the largest possible hyperopic cop number. We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number. We show that for planar graphs, as with the usual cop number, the hyperopic cop number is at most 3. The hyperopic cop number is considered for countable graphs, and it is shown that for connected chains of graphs, the hyperopic cop density can be any real number in $[0,1/2].$