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Zusammenfassung:
{
We analyze a randomized pursuit-evasion game on graphs. This game is
played by two players, a {\em hunter} and a {\em rabbit}. Let
$G$ be any connected, undirected graph with $n$ nodes.
The game is played in rounds and in each round both the
hunter and the rabbit are located at a node of the graph. Between rounds
both the hunter and the rabbit can stay at the current node or move to
another node. The hunter is assumed to be {\em restricted} to the
graph $G$: in every round, the hunter can move using at most one edge.
For the rabbit we investigate two models: in one
model the rabbit is restricted to the same graph as the hunter, and in
the other model the rabbit is {\em unrestricted}, i.e., it can jump to
an arbitrary node in every round.
We say that the rabbit is {\em caught}\/ as soon as hunter and rabbit
are located at the same node in a round. The goal of the hunter is to
catch the rabbit in as few rounds as possible, whereas the rabbit aims
to maximize the number of rounds until it is caught. Given a
randomized hunter strategy for $G$, the {\em escape length} for that
strategy is the worst case expected number of rounds it takes the
hunter to catch the rabbit, where the worst case is with regards to
all (possibly randomized) rabbit strategies.
Our main result is a hunter strategy for general graphs
with an escape length of only $\O(n \log (\diam(G)))$ against
restricted as well as unrestricted rabbits.
This bound is close to optimal since $\Omega(n)$ is a trivial lower bound
on the escape length in both models.
Furthermore, we prove that our upper bound is optimal
up to constant factors against unrestricted rabbits.
}
