Given a hypergraph $H = (V, E)$, call $H' = (V, E')$ a refinement of $H$ iff there exists a partition $p : E' \to I$ (where $I$ is an arbitrary index set) such that $E = \{\bigcup_{x \in p^{-1}(i)} x \mid i \in I\}$. Essentially this means that there's some way to partition the hyperedges in $E'$ such that merging together all the hyperedges in each part of the partition gives $E$. Note that this isn't quite the same as just splitting the edges in $E$, $(\{a, b, c\}, \{\{a, b\}, \{b, c\}\})$ would be a valid refinement of $(\{a, b, c\}, \{\{a, b, c\}\})$.
I made up this terminology, is there a name for this in the literature? It seems like a natural concept.
I'm particularly interested in solving a minimization problem where a weighted (i.e.: equipped with a function $w: E' \to \mathbb{N}$) hypergraph refinement is the variable, and the set of nodes in the hypergraph is weighted by a function $c: V \to \mathbb{N}$ that we'll call the capacity of a node. The only constraint is that the sum of the weights of the hyperedges incident on a given node must be less than or equal to its capacity. The objective function is $|E'| + |\{v \in V \mid L(v) > 0\}|$ where $L(v)$ is the "leftover weight", the difference between the capacity of $v$ and the sum of incident hyperedge weights, i.e.: $L(v) = c(v) - \Sigma_{e \in E', v \in e} w(e)$. The constraint is then formally stated as $L(v) \ge 0$ for all $v \in V$. Intuitively, this is like a combination of a minimum cover problem with a subset-sum on each node.
Is any of this secretly equivalent to something else?
For context, I'm trying to solve this problem where the hypergraph came from the maximal cliques of a graph (i.e.: we're really trying to find weights for each clique in a graph such that the sum of the incident cliques' weights is less than a per-node constant).
