I have a set of sets $S=\{s_{1}, s_{2} \ldots s_{n}\}$ that I want to transform into a different set of sets $T=\{t_{1}, t_{2} \ldots t_{m}\}$, $\forall n,m$, where:
$\sum_{i=1}^{n}|s_{i}|=\sum_{j=1}^{m}|t_{j}|$,
$\forall i \ne j \ \ s_{i}\cap s_{j} = \emptyset , \ t_{i}\cap t_{j} = \emptyset$
$|t_{j}|_{j=1}^{m}$ is a given
and $\forall t$, elements are sourced from $\{s_{1}\cup s_{2} \cup \ldots \cup s_n\}$
I need to find a method/algorithm that produces $T$ with minimum dispersion of $s$ elements, ie, I need to keep $s_i$ elements as together as possible in $t_j$, say minimize $\sum_{i,j} [s_{i}\cap t_{j}\neq\emptyset]$ (for which I mean the count of all non empty intersections of elements from $S$ and $T$.)
I've tried to figure this out but currently I am at a loss. Any pointers to literature or a possible approach is most welcome.
TIA, Luis
Say you have only one set $t$ and there are duplicate elements in the $s_{i}$: $\left|\cup_{i}s_{i}\right|<\sum\left|s_{i}\right|$ and $\left|t\right|\overset{!}{=}\sum\left|s_{i}\right|$. Your problem is not well posed in this case, as per the standard definition of a set, $t$ cannot contain any duplicates. You need to rethink your whole situation.