  In the NP-hard continuous 1.5D Terrain Guarding Problem (TGP) we are given an
$x$-monotone chain of line segments in $\mathbb{R}^2$ (the terrain $T$) and ask
for the minimum number of guards (located anywhere on $T$) required to guard
all of $T$. We construct guard candidate and witness sets $G, W \subset T$ of
polynomial size such that any feasible (optimal) guard cover $G^* \subseteq G$
for $W$ is also feasible (optimal) for the continuous TGP. This discretization
allows us to (1) settle NP-completeness for the continuous TGP, (2) provide a
Polynomial Time Approximation Scheme (PTAS) for the continuous TGP using the
PTAS for the discrete TGP by Gibson et al., and (3) formulate the continuous
TGP as an Integer Linear Program (IP). Furthermore, we propose several
filtering techniques reducing the size of our discretization, allowing us to
devise an efficient IP-based algorithm that reliably provides optimal guard
placements for terrains with up to $10^6$ vertices within minutes on a standard
desktop computer.
