# Local edge-connectivity augmentation of a hypergraph with fixed rank

We are given a hypergraph $G_0=(V,\mathcal{E}_0)$ of rank at most $k$ (where $k$ is fixed, not part of the input) and a symmetric function $r:V\times V\to \mathbb{Z}_+$. Can we find in polynomial time a graph $G=(V,{E})$ with a minimum number of edges such that $\lambda_{G_0+G}(x,y)\ge r(x,y)$ for every $x,y\in V$?
The rank of a hypergraph is the size of its largest hyperedge. The problem is solved for k=2 (i.e. when $G_0$ is a graph) by Frank [1], and this implies a solution for k=3 because a 3-hypergraph can be transformed into a graph with the same local edge-connectivities and vice versa [2]. The first open case is k=4. If k is not fixed, then the problem is NP-complete [3].