  Let $P$ be a set of $n$ points in the plane. We consider a variation of the
classical Erd\H{o}s-Szekeres problem, presenting efficient algorithms with
$O(n^3)$ running time and $O(n^2)$ space complexity that compute: (1) A subset
$S$ of $P$ such that the boundary of the rectilinear convex hull of $S$ has the
maximum number of points from $P$, (2) a subset $S$ of $P$ such that the
boundary of the rectilinear convex hull of $S$ has the maximum number of points
from $P$ and its interior contains no element of $P$, (3) a subset $S$ of $P$
such that the rectilinear convex hull of $S$ has maximum area and its interior
contains no element of $P$, and (4) when each point of $P$ is assigned a
weight, positive or negative, a subset $S$ of $P$ that maximizes the total
weight of the points in the rectilinear convex hull of $S$.
