  We present algorithms for the Max-Cover and Max-Unique-Cover problems in the
data stream model. The input to both problems are $m$ subsets of a universe of
size $n$ and a value $k\in [m]$. In Max-Cover, the problem is to find a
collection of at most $k$ sets such that the number of elements covered by at
least one set is maximized. In Max-Unique-Cover, the problem is to find a
collection of at most $k$ sets such that the number of elements covered by
exactly one set is maximized. Our goal is to design single-pass algorithms that
use space that is sublinear in the input size. Our main algorithmic results
are:
  If the sets have size at most $d$, there exist single-pass algorithms using
$\tilde{O}(d^{d+1} k^d)$ space that solve both problems exactly. This is
optimal up to polylogarithmic factors for constant $d$.
  If each element appears in at most $r$ sets, we present single pass
algorithms using $\tilde{O}(k^2 r/\epsilon^3)$ space that return a $1+\epsilon$
approximation in the case of Max-Cover. We also present a single-pass algorithm
using slightly more memory, i.e., $\tilde{O}(k^3 r/\epsilon^{4})$ space, that
$1+\epsilon$ approximates Max-Unique-Cover.
  In contrast to the above results, when $d$ and $r$ are arbitrary, any
constant pass $1+\epsilon$ approximation algorithm for either problem requires
$\Omega(\epsilon^{-2}m)$ space but a single pass $O(\epsilon^{-2}mk)$ space
algorithm exists. In fact any constant-pass algorithm with an approximation
better than $e/(e-1)$ and $e^{1-1/k}$ for Max-Cover and Max-Unique-Cover
respectively requires $\Omega(m/k^2)$ space when $d$ and $r$ are unrestricted.
En route, we also obtain an algorithm for a parameterized version of the
streaming Set-Cover problem.
