1,803 research outputs found
Cops and Robbers is EXPTIME-complete
We investigate the computational complexity of deciding whether k cops can
capture a robber on a graph G. In 1995, Goldstein and Reingold conjectured that
the problem is EXPTIME-complete when both G and k are part of the input; we
prove this conjecture.Comment: v2: updated figures and slightly clarified some minor point
Cops and Invisible Robbers: the Cost of Drunkenness
We examine a version of the Cops and Robber (CR) game in which the robber is
invisible, i.e., the cops do not know his location until they capture him.
Apparently this game (CiR) has received little attention in the CR literature.
We examine two variants: in the first the robber is adversarial (he actively
tries to avoid capture); in the second he is drunk (he performs a random walk).
Our goal in this paper is to study the invisible Cost of Drunkenness (iCOD),
which is defined as the ratio ct_i(G)/dct_i(G), with ct_i(G) and dct_i(G) being
the expected capture times in the adversarial and drunk CiR variants,
respectively. We show that these capture times are well defined, using game
theory for the adversarial case and partially observable Markov decision
processes (POMDP) for the drunk case. We give exact asymptotic values of iCOD
for several special graph families such as -regular trees, give some bounds
for grids, and provide general upper and lower bounds for general classes of
graphs. We also give an infinite family of graphs showing that iCOD can be
arbitrarily close to any value in [2,infinty). Finally, we briefly examine one
more CiR variant, in which the robber is invisible and "infinitely fast"; we
argue that this variant is significantly different from the Graph Search game,
despite several similarities between the two games
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
A probabilistic version of the game of Zombies and Survivors on graphs
We consider a new probabilistic graph searching game played on graphs,
inspired by the familiar game of Cops and Robbers. In Zombies and Survivors, a
set of zombies attempts to eat a lone survivor loose on a given graph. The
zombies randomly choose their initial location, and during the course of the
game, move directly toward the survivor. At each round, they move to the
neighbouring vertex that minimizes the distance to the survivor; if there is
more than one such vertex, then they choose one uniformly at random. The
survivor attempts to escape from the zombies by moving to a neighbouring vertex
or staying on his current vertex. The zombies win if eventually one of them
eats the survivor by landing on their vertex; otherwise, the survivor wins. The
zombie number of a graph is the minimum number of zombies needed to play such
that the probability that they win is strictly greater than 1/2. We present
asymptotic results for the zombie numbers of several graph families, such as
cycles, hypercubes, incidence graphs of projective planes, and Cartesian and
toroidal grids
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