Let $S_1,\ldots,S_m\subseteq U$ be subsets of a set $U$ of size $\lvert U\rvert=n$. Over all permutations $\pi$ of the set $\\{1,\ldots,m\\}$, I want to maximize  the quantity\begin{equation}\sum_{k=1}^m\left\lvert\bigcup_{i=1}^k S_{\pi(i)} \right\rvert.\end{equation}This problem is NP-complete as it contains Exact cover by 3-sets as a special case: Suppose $n$ is divisible by 3, say $n=3n'$, and all sets $S_i$ have size 3. Then the upper bound\begin{equation}3(1+2+\cdots+n')+(m-n')n\end{equation}can be achieved if and only if $U$ can be covered by $n'$ of the 3-sets. I'm interested in the hardness of approximation. For instance, can the hardness of approximation for set cover be used to say something about the above problem? 
Approximation for accumulative set cover
approximation algorithms;approximation hardness