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A family of sets $F = \{S_1, \dots, S_n\}$ on the ground set $S$ is laminar, if for every $1\leq i < j \leq n$, either $S_i \subsetneq S_j$ or $S_j \subsetneq S_i$ or $S_i \cap S_j = \varnothing$ and also for every $1 \leq i \leq n: \varnothing \neq S_i \subseteq S$.

The question is given a family of sets $F = \{S_1, \dots, S_n\}$, find laminar family $F' = \{S'_1, \dots, S'_n\}$ so that for each i we have $S_i \subseteq S'_i$ and $\sum_i |S'_i - S_i|$ is minimized. In other words we want to find the smallest extension of $F$ which is laminar.

Note that there may be no such extension for $F$. For example if for $i\neq j$ we have $S_i = S_j = S$ then there exist no laminar extension for $F$. Is there any efficient algorithm for finding the smallest laminar extension of $F$?

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