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Given $m$ sets $S_1, S_2, \dots, S_m$ and a bound $b$, find as many sets as possible among $m$ sets, says $S_{i_i}, S_{i_2}, \dots, S_{i_k}$ such that

$$\big| S_{i_i} \cup S_{i_2} \cup \cdots \cup S_{i_k} \big| < b$$ I came across the problem above in computer science research. Since my mathematics background is so-so, I would really appreciate any hints.

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    $\begingroup$ "largest $k$ sets" is ambiguous, please edit to make it clearer. Also, is it an algorithm you are asking for? $\endgroup$ – Brendan McKay Aug 6 '18 at 6:57
  • $\begingroup$ @BrendanMcKay I have updated the question, hope it is clearer now. I'm looking for an algorithm to find these $k$ sets. Or better, a proof saying this is NP-hard since this problem looks pretty familiar to a set cover problem. Regards, $\endgroup$ – Tran Muoi Aug 6 '18 at 8:26

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