Say I have some collection of ordered sets $C = \{S_i\}$. Is there an efficient way to determine the minimal ordered set $S^*$ such that each of the original $S_i \subseteq S^*$, with order preserved?
Simple example: $C = \{\{a, b, c\}, \{b, d, f\}, \{b, c, d\}\}$, then $S^* = \{a, b, c, d, f\}$
More complex example: $C = \{\{a, b, c\}, \{b, c, d\}, \{x, a, d\}, \{b, b, a, c\}, \{y, b, c, b\}\}$, then one solution is $S^* = \{y, x, a, b, c, b, a, c, d \}$.
Another example: $C = \{\{a, b, c\}, \{b, a, c\}, \{c, a, b\}, \{b, c, a\}, \{c, b, c\}, \{b, b, c\}\}$, $S^* = \{b, c, a, b, c\}$
The naïve solution is to just concatenate the $S_i$. When $\cap S_i = \emptyset$, then of course this happens to also be the minimal solution.
Perhaps finding some traversal of a digraph representing all elements and their ordering?
I've found out that this is known as the shortest common supersequence problem. Research and papers on the topic can be found online using this search term.