2,868 research outputs found
Variations on Cops and Robbers
We consider several variants of the classical Cops and Robbers game. We treat
the version where the robber can move R > 1 edges at a time, establishing a
general upper bound of N / \alpha ^{(1-o(1))\sqrt{log_\alpha N}}, where \alpha
= 1 + 1/R, thus generalizing the best known upper bound for the classical case
R = 1 due to Lu and Peng. We also show that in this case, the cop number of an
N-vertex graph can be as large as N^{1 - 1/(R-2)} for finite R, but linear in N
if R is infinite. For R = 1, we study the directed graph version of the
problem, and show that the cop number of any strongly connected digraph on N
vertices is at most O(N(log log N)^2/log N). Our approach is based on
expansion.Comment: 18 page
Jeux de policiers et voleurs : modĂšles et applications
Les jeux de policiers et voleurs sont Ă©tudiĂ©s depuis une trentaine dâannĂ©es en informatique et en mathĂ©matiques. Comme dans les jeux de poursuite en gĂ©nĂ©ral, des poursuivants (les policiers) cherchent Ă capturer des Ă©vadĂ©s (les voleurs), cependant ici les joueurs agissent tour Ă tour et sont contraints de se dĂ©placer sur une structure discrĂšte. On suppose toujours que les joueurs connaissent les positions exactes de leurs opposants, autrement dit le jeu se dĂ©roule Ă information parfaite. La premiĂšre dĂ©finition dâun jeu de policiers-voleurs remonte Ă celle de Nowakowski et Winkler [39] et, indĂ©pendamment, Quilliot [46]. Cette premiĂšre dĂ©finition prĂ©sente un jeu opposant un seul policier et un seul voleur avec des contraintes sur leurs vitesses de dĂ©placement. Des extensions furent graduellement proposĂ©es telles que lâajout de policiers et lâaugmentation des vitesses de mouvement. En 2014, Bonato et MacGillivray [6] proposĂšrent une gĂ©nĂ©ralisation des jeux de policiers-voleurs pour permettre lâĂ©tude de ceux-ci dans leur globalitĂ©. Cependant, leur modĂšle ne couvre aucunement les jeux possĂ©dant des composantes stochastiques tels que ceux dans lesquels les voleurs peuvent bouger de maniĂšre alĂ©atoire. Dans ce mĂ©moire est donc prĂ©sentĂ© un nouveau modĂšle incluant des aspects stochastiques. En second lieu, on prĂ©sente dans ce mĂ©moire une application concrĂšte de lâutilisation de ces jeux sous la forme dâune mĂ©thode de rĂ©solution dâun problĂšme provenant de la thĂ©orie de la recherche. Alors que les jeux de policiers et voleurs utilisent lâhypothĂšse de lâinformation parfaite, les problĂšmes de recherches ne peuvent faire cette supposition. Il appert cependant que le jeu de policiers et voleurs peut ĂȘtre analysĂ© comme une relaxation de contraintes dâun problĂšme de recherche. Ce nouvel angle de vue est exploitĂ© pour la conception dâune borne supĂ©rieure sur la fonction objectif dâun problĂšme de recherche pouvant ĂȘtre mise Ă contribution dans une mĂ©thode dite de branch and bound.Cops and robbers games have been studied for the last thirty years in computer science and mathematics. As in general pursuit evasion games, pursuers (cops) seek to capture evaders (robbers), however here the players move in turn and are constrained to move on a discrete structure. It is always assumed that players know the exact location of their adversary, in other words the game is played with perfect information. The first definition of a cops and robbers game dates back to Nowakowski and Winkler [39] and, independantly, Quilliot [46]. This first definition presents a game opposing a single cop against a lone robber, both with constraints on their speed. Extensions were gradually formulated such as increasing the number of cops and the speed of the players. In 2014, Bonato and MacGillivray [6] presented a general characterization of cops and robbers games in order for them to be globally studied. However, their model does not take into account stochastic events that may occur such as the robbers moving in a random fashion. In this thesis, a novel model that includes stochastic elements is presented. Furthermore, we present in this thesis a concrete application of cops and robbers games in the form of a method of resolution of a problem from search theory. Although cops and robbers games assume perfect information, this hypothesis cannot be maintained in search problems. It appears however that cops and robbers games can be viewed as constraint relaxations of search problems. This point of view is made use of in the conception of an upper bound on the objective function of a search problem that is a applied in a branch and bound method
To catch a falling robber
We consider a Cops-and-Robber game played on the subsets of an -set. The
robber starts at the full set; the cops start at the empty set. On each turn,
the robber moves down one level by discarding an element, and each cop moves up
one level by gaining an element. The question is how many cops are needed to
ensure catching the robber when the robber reaches the middle level. Aaron Hill
posed the problem and provided a lower bound of for even and
for odd . We prove an
upper bound (for all ) that is within a factor of times this
lower bound.Comment: Minor revision
Bounds on the length of a game of Cops and Robbers
In the game of Cops and Robbers, a team of cops attempts to capture a robber on a graph G. All players occupy vertices of G. The game operates in rounds; in each round the cops move to neighboring vertices, after which the robber does the same. The minimum number of cops needed to guarantee capture of a robber on G is the cop number of G, denoted c(G), and the minimum number of rounds needed for them to do so is the capture time. It has long been known that the capture time of an n-vertex graph with cop number k is O(nk+1). More recently, Bonato, Golovach, Hahn, and KratochvĂl ([3], 2009) and GavenÄiak ([10], 2010) showed that for k = 1, this upper bound is not asymptotically tight: for graphs with cop number 1, the cop can always win within n â 4 rounds. In this paper, we show that the upper bound is tight when k â„ 2: for fixed k â„ 2, we construct arbitrarily large graphs G having capture time at least (|V (G)|/40k4 )k+1.
In the process of proving our main result, we establish results that may be of independent interest. In particular, we show that the problem of deciding whether k cops can capture a robber on a directed graph is polynomial-time equivalent to deciding whether k cops can capture a robber on an undirected graph. As a corollary of this fact, we obtain a relatively short proof of a major conjecture of Goldstein and Reingold ([11], 1995), which was recently proved through other means ([12], 2015). We also show that n-vertex strongly-connected directed graphs with cop number 1 can have capture time Ω(n2), thereby showing that the result of Bonato et al. [3] does not extend to the directed setting
Visibility Graphs, Dismantlability, and the Cops and Robbers Game
We study versions of cop and robber pursuit-evasion games on the visibility
graphs of polygons, and inside polygons with straight and curved sides. Each
player has full information about the other player's location, players take
turns, and the robber is captured when the cop arrives at the same point as the
robber. In visibility graphs we show the cop can always win because visibility
graphs are dismantlable, which is interesting as one of the few results
relating visibility graphs to other known graph classes. We extend this to show
that the cop wins games in which players move along straight line segments
inside any polygon and, more generally, inside any simply connected planar
region with a reasonable boundary. Essentially, our problem is a type of
pursuit-evasion using the link metric rather than the Euclidean metric, and our
result provides an interesting class of infinite cop-win graphs.Comment: 23 page
Chasing robbers on random geometric graphs---an alternative approach
We study the vertex pursuit game of \emph{Cops and Robbers}, in which cops
try to capture a robber on the vertices of the graph. The minimum number of
cops required to win on a given graph is called the cop number of . We
focus on , a random geometric graph in which vertices are
chosen uniformly at random and independently from , and two vertices
are adjacent if the Euclidean distance between them is at most . The main
result is that if then the cop number is
with probability that tends to as tends to infinity. The case was
proved earlier and independently in \cite{bdfm}, using a different approach.
Our method provides a tight upper bound for the number of rounds
needed to catch the robber.Comment: 6 page
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