Connected Surveillance Game

International audienceThe surveillance game [Fomin et al., 2012] models the problem of web-page prefetching as a pursuit evasion game played on a graph. This two-player game is played turn-by-turn. The first player, called the observer, can mark a fixed amount of vertices at each turn. The second one controls a surfer that stands at vertices of the graph and can slide along edges. The surfer starts at some initially marked vertex of the graph, its objective is to reach an unmarked node before all nodes of the graph are marked. The surveillance number sn(G) of a graph G is the minimum amount of nodes that the observer has to mark at each turn ensuring it wins against any surfer in G. Fomin et al. also defined the connected surveillance game where the observer must ensure that marked nodes always induce a connected subgraph. They ask what is the cost of connectivity, i.e., is there a constant c > 0 such that the ratio between the connected surveillance number csn(G) and sn(G) is at most c for any graph G. It is straightforward to show that csn(G) ≤ ∆ sn(G) for any graph G with maximum degree ∆. Moreover, it has been shown that there are graphs G for which csn(G) = sn(G) + 1. In this paper, we investigate the question of the cost of the connectivity. We first provide new non-trivial upper and lower bounds for the cost of connectivity in the surveillance game. More precisely, we present a family of graphs G such that csn(G) > sn(G) + 1. Moreover, we prove that csn(G) ≤ sn(G)n for any n-node graph G. While the gap between these bounds remains huge, it seems difficult to reduce it. We then define the online surveillance game where the observer has no a priori knowledge of the graph topology and discovers it little-by-little. This variant, which fits better the prefetching motivation, is a restriction of the connected variant. Unfortunately, we show that no algorithm for solving the online surveillance game has competitive ratio better than Ω(∆). That is, while interesting, this variant does not help to obtain better upper bounds for the connected variant. We finally answer an open question [Fomin et al., 2012] by proving that deciding if the surveillance number of a digraph with maximum degree 6 is at most 2 is NP-hard

Connected Surveillance Game

International audienceThe surveillance game [Fomin et al., 2012] models the prob- lem of web-page prefetching as a pursuit evasion game played on a graph. This two-player game is played turn-by-turn. The rst player, called the observer, can mark a xed amount of vertices at each turn. The second one controls a surfer that stands at vertices of the graph and can slide along edges. The surfer starts at some initially marked vertex of the graph, her objective is to reach an unmarked node The surveillance number sn(G) of a graph G is the minimum amount of nodes that the observer has to mark at each turn ensuring it wins against any surfer in G. Fomin et al. also de ned the connected surveillance game where the marked nodes must always induce a connected subgraph. They ask if there is a constant c > 0 such that csn(G)/ sn(G) sn(G)+1. Moreover, we prove that csn(G) <= sn(G) n^(1/2) for any n-node graph G. While the gap between these bounds remains huge, it seems di cult to reduce it. We then de ne the online surveillance game where the observer has no a priori knowledge of the graph topology and discovers it little-by- little. Unfortunately, we show that no algorithm for solving the online surveillance game has competitive ratio better than Omega(Delta)

A Connected Version of the Graph Coloring Game

The graph coloring game is a two-player game in which, given a graph G and a set of k colors, the two players, Alice and Bob, take turns coloring properly an uncolored vertex of G, Alice having the first move. Alice wins the game if and only if all the vertices of G are eventually colored. The game chromatic number of a graph G is then defined as the smallest integer k for which Alice has a winning strategy when playing the graph coloring game on G with k colors. In this paper, we introduce and study a new version of the graph coloring game by requiring that, after each player's turn, the subgraph induced by the set of colored vertices is connected. The connected game chromatic number of a graph G is then the smallest integer k for which Alice has a winning strategy when playing the connected graph coloring game on G with k colors. We prove that the connected game chromatic number of every outerplanar graph is at most 5 and that there exist outerplanar graphs with connected game chromatic number 4. Moreover, we prove that for every integer k ≥ 3, there exist bipartite graphs on which Bob wins the connected coloring game with k colors, while Alice wins the connected coloring game with two colors on every bipartite graph

Understanding the Power of Stigmergy of Anonymous Agents in Discrete Environments

International audienceCommunication by stigmergy consists, for agents/robots devoid of other dedicated communication devices, in exchanging information by observing each other's movements, similar to how honeybees use a dance to inform each other on the location of food sources. Stigmergy, while a popular technique in soft computing (e.g., swarm intelligence and swarm robotics), has received little attention from a computational viewpoint, with only one study proposing a method in a continuous environment. An important question is whether there are limits intrinsic to the environment on the feasibility of stigmergy. While it is not the case in a continuous environment, we show that the answer is quite different when the environment is discrete. This paper considers stigmergy in graphs and identifies classes of graphs in which robots can communicate by stigmergy. We provide two algorithms with different tradeoffs. One algorithm achieves faster stigmergy when the density of robots is low enough to let robots move independently. This algorithm works when the graph contains some particular pairwise-disjoint subgraphs. The second algorithm, while slower solves the problem under an extremely high density of robots assuming that the graph admits some large cycle. Both algorithms are described in a general way, for any graph that admits the desired properties and with identified nodes. We show how the latter assumption can be removed in more specific topologies. Indeed, we consider stigmergy in the grid which offers additional orientation information not available in a general graphs, allowing us to relax some of the assumptions. Given an $N\times M$ anonymous grid, we show that the first algorithm requires $O(\mathcal{M})$ steps to achieve communication by stigmergy, where $\mathcal{M}$ is the maximum length of a communication message, but it works only if the number of robots is less than $\left\lfloor\frac{N\cdot M}{9}\right\rfloor$. The second algorithm, which requires $O(k^2)$ steps, where $k$ is the number of robots, on the other hand, works for up to $N\cdot M-5$ robots. In both cases, we consider very weak assumptions on the robots capabilities: i.e., we assume that the robots are anonymous, asynchronous, uniform, and execute deterministic algorithms

Analysis of prefetching methods from a graph-theoretical perspective

Είναι σημαντικό να τονίσουμε το ρόλο που τα Δίκτυα Διανομής Περιεχομένου (CDNs) παίζουν στις ταχέως αναπτυσσόμενες τοπολογίες του Διαδικτύου. Είναι υπεύθυνα για την εξυπηρέτηση της πλειοψηφίας του περιεχομένου του Διαδικτύου στους τελικούς χρήστες αντιγράφοντας το από το διακομιστή προέλευσης και τοποθετώντας το σε έναν διακομιστή πιο κοντά τους. Τα μεγαλύτερα ίσως προβλήματα που αντιμετωπίζουν τα CDNs έχουν να κάνουν με την επιλογή του περιεχομένου που πρέπει να προανακτηθεί αλλά και την επιλογή ενός κατάλληλου διακομιστή μεσολάβησης στον οποίο θα τοποθετηθεί. Εμείς θα επικεντρωθούμε στο πρόβλημα προανάκτησής περιεχομένου επεκτείνοντας την έρευνα που έγινε από τον Σιδηρόπουλο κ.α. (WorldWideWebJournal, vol. 11, 2008, pp. 39-70). Συγκεκριμένα, θα προσπαθήσουμε να αποφανθούμε πώς η μέθοδος συσταδοποίησής τους μπορεί να δουλέψει σε συγκεκριμένα περιβάλλοντα σε σύγκριση με μια άλλη προσέγγιση που χρησιμοποιείται για την επίλυση του παιχνιδιού επιτήρησης σε γράφους όπως διερευνήθηκε από τον Fomin κ.α. (Proc. 6thInt’lConf. onFUNwithAlgorithms, 2012, pp.166-176) και τον Giroire κ.α. . (JournalofTheoreticalComputerScience, vol. 584, 2015, pp.131-143). Στην πορεία, δίνουμε και έναν άλλο ορισμό για τη συνοχή των συστάδων που καλύπτει και οριακές περιπτώσεις. Τέλος, ορίζουμε ένα καινούριο πρόβλημα, τη διαμέριση δηλαδή του γράφου σε έναν προκαθορισμένο αριθμό ανεξάρτητων συστάδων με βέλτιστη μέση συνοχή.It is important to highlight the role Content Distribution Networks (CDNs) play in rapidly growing Internet topologies. They are responsible for serving the lion&apos;s share of Internet content to the end users by replicating it from the origin server and placing it to a caching server closer to them. Probably the biggest issues CDNs have to deal with revolve around deciding which content gets prefetched, in which surrogate/caching server it is placed and allocating storage to each server in an efficient manner. We will focus on the content selection/prefetching problem extending the work done by Sidiropoulos et al. (World Wide Web Journal, vol. 11, 2008, pp. 39-70). Specifically, we are trying to determine how their clustering algorithm can work in specific environments in comparison with an approach used to solve the surveillance game in graphs as discussed by Fomin et al(Proc. 6th Int’l Conf. on FUN with Algorithms, 2012, pp.166-176)and Giroire et al. (Journal of Theoretical Computer Science, vol. 584, 2015, pp.131-143). Along the way, we provide another definition for cluster cohesion that accounts for edge cases. Finally, we define an original problem, which consists of partitioning a graph into a predefined amount of disjoint clusters of optimal average cohesion

Perpetually Dominating Large Grids

In the Eternal Domination game, a team of guard tokens initially occupies a dominating set on a graph G. A rioter then picks a node without a guard on it and attacks it. The guards defend against the attack: one of them has to move to the attacked node, while each remaining one can choose to move to one of his neighboring nodes. The new guards' placement must again be dominating. This attack-defend procedure continues perpetually. The guards win if they can eternally maintain a dominating set against any sequence of attacks, otherwise the rioter wins. We study rectangular grids and provide the first known general upper bound for these graphs. Our novel strategy implements a square rotation principle and eternally dominates m x n grids by using approximately (mn)/5 guards, which is asymptotically optimal even for ordinary domination

Eternally dominating large grids

© 2018 Elsevier B.V. In the m-Eternal Domination game, a team of guard tokens initially occupies a dominating set on a graph G. An attacker then picks a vertex without a guard on it and attacks it. The guards defend against the attack: one of them has to move to the attacked vertex, while each remaining one can choose to move to one of his neighboring vertices. The new guards’ placement must again be dominating. This attack-defend procedure continues eternally. The guards win if they can eternally maintain a dominating set against any sequence of attacks, otherwise the attacker wins. The m-eternal domination number for a graph G is the minimum amount of guards such that they win against any attacker strategy in G (all guards move model). We study rectangular grids and provide the first known general upper bound on the m-eternal domination number for these graphs. Our novel strategy implements a square rotation principle and eternally dominates m×n grids by using approximately [Formula presented] guards, which is asymptotically optimal even for ordinary domination

Optimal Prefetching in Random Trees

International audienceWe propose and analyze a model for optimizing the prefetching of documents, in the situation where the connection between documents is discovered progressively. A random surfer moves along the edges of a random tree representing possible sequences of documents, which is known to a controller only up to depth d. A quantity k of documents can be prefetched between two movements. The question is to determine which nodes of the known tree should be prefetched so as to minimize the probability of the surfer moving to a node not prefetched. We analyzed the model with the tools of Markov decision process theory. We formally identified the optimal policy in several situations, and we identified it numerically in others