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The greedy algorithm gives a $1-1/e$ approximation for maximum coverage subject to a cardinality constraint, and this known to be optimal (unless P=NP). The same algorithm also gives a $1-1/e$ approximation for the more general problem of monotone submodular maximization subject to a cardinality constraint, and this is also known to be optimal in the value oracle model.

What happens when the cardinality constraint $k$ is a fixed fraction $c$ of the total number of sets $n$, that is, $k = cn$? A random solution gives a $c$ approximation, which already improves on $1-1/e$ when $c > 1-1/e$. We show that for every $c > 0$, both approximation algorithms can be improved.

In the case of monotone submodular maximization subject to a cardinality constraint, we show that the measured continuous greedy algorithm gives a $1-(1-c)^{1/c}$ approximation, which is tight when $1/c$ is an integer; we conjecture it to be tight for all $c$.

In the case of maximum coverage subject to a cardinality constraint, we first show that the natural LP gives a $1-(1-c)^{1/c}$ approximation when $1/c$ is an integer and a better approximation when $1/c$ is fractional. We give an integrality gap which matches our rounding scheme. The integrality gap also translates to a value oracle hardness for monotone submodular maximization subject to a cardinality constraint.

We then show how to improve on the LP using an SDP for $c = 1/2$ (provably) and for $1/2 < c < 1$ (conjecturally and numerically). This separates the two problems, showing that maximum coverage is more approximable than monotone submodular maximization in this setting. To the best of our knowledge, this is the first such separation in a natural setting.