  An ever-important issue is protecting infrastructure and other valuable
targets from a range of threats from vandalism to theft to piracy to terrorism.
The "defender" can rarely afford the needed resources for a 100% protection.
Thus, the key question is, how to provide the best protection using the limited
available resources. We study a practically important class of security games
that is played out in space and time, with targets and "patrols" moving on a
real line. A central open question here is whether the Nash equilibrium (i.e.,
the minimax strategy of the defender) can be computed in polynomial time. We
resolve this question in the affirmative. Our algorithm runs in time polynomial
in the input size, and only polylogarithmic in the number of possible patrol
locations (M). Further, we provide a continuous extension in which patrol
locations can take arbitrary real values. Prior work obtained polynomial-time
algorithms only under a substantial assumption, e.g., a constant number of
rounds. Further, all these algorithms have running times polynomial in M, which
can be very large.
