Given a set $N = \{1, 2, \dotsc,n\}$, let $F \subseteq 2^N$ represent a family of subsets of $N$. Each subset $S \in F$ has a reward $+1$ or $-1$. We seek to select a subset $K \subseteq N$ such that the total "collected" rewards of sets $S \in F$ is maximized. The reward corresponding to a set $S \in F$ is "collected" if and only if $K \subseteq S$.
Example: Let $N = \{1, 2, 3, 4, 5\}$. Define $S_1 = \{1, 2, 5\}$, $S_2 = \{2, 4\}$ and $S_3 = \{1, 2, 3\}$ with weights $+1$. Also, define $S_4 = \{2\}$, $S_5 = \{2, 4, 5\}$, $S_6 = \{1, 4\}$ with weights $-1$. The optimal selection is $K = \{1, 2\}$ as it can collect the $+1$ reward for both $S_1$ and $S_3$. If we choose $K = \{2\}$, while we collect the $+1$ reward for $S_1$, $S_2$ and $S_3$, we also collect the $-1$ reward for $S_4$ and $S_5$.
Is there any well-known combinatorial problem with a structure similar to the above problem? In particular, one that can be reduced to this problem for complexity analysis.