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I've encountered some problem which seems general enough to have already been solved.

There is a set of objects $O=\{o_1, o_2,\dots,o_k\}$ and a family of sets $A_1,A_2,\dots,A_t \subseteq O$.

For every $1 \leq j < k$ we need to find a subset $O' \subset O$ of size $j$ maximizing the number of sets $A_i$ contained in it.

What is the best algorithm to solve this? I've been thinking about reducing it to a problem in graphs or flow networks but still haven't arrived at a solution.

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Your problem is essentially Minimum $k$-Union. In this problem (switching from $k$ to $\ell$), you want to find $\ell$ sets out of $A_1,\ldots,A_t$ which together cover the least number of elements. Denoting by $f(\ell)$ the solution of this problem and by $g(j)$ the solution of your problem, we have $$ f(\ell) \leq j \Longleftrightarrow \ell \leq g(j). $$ In other words, $g(j)$ is the maximal $\ell$ such that $f(\ell) \leq j$.

Unfortunately, Minimum $k$-Union is NP-hard. You can find some links in this question on StackOverflow.

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  • $\begingroup$ Thanks for the reply. I see the similarity, the problem is I want to find the solution for every $1 \leq j < k$. I'd like to use an approximated solution to the minimum k-union, but to solve for every j I'd need to apply the solution for $ 1,2,...,numof matches=M $. since $M$ can be as large as $2^k$ that's not very good. If the processing time of the solution is $T$ then a total time of $2^k * T$ will be needed, whereas if there is a solution which its input is the number of objects from $O$ and which takes time $R$, a total time of $k*R$ is needed, assuming $T \approx R$ this is better. $\endgroup$ – Gilad Deutsch Apr 1 '20 at 18:36

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