  A graph $G=(V,E)$ is a support of a hypergraph $H=(V,S)$ if every hyperedge
induces a connected subgraph in $G$. Supports are used for certain types of
hypergraph visualizations. In this paper we consider visualizing spatial
hypergraphs, where each vertex has a fixed location in the plane. This is the
case, e.g., when modeling set systems of geospatial locations as hypergraphs.
By applying established aesthetic quality criteria we are interested in finding
supports that yield plane straight-line drawings with minimum total edge length
on the input point set $V$. We first show, from a theoretical point of view,
that the problem is NP-hard already under rather mild conditions as well as a
negative approximability results. Therefore, the main focus of the paper lies
on practical heuristic algorithms as well as an exact, ILP-based approach for
computing short plane supports. We report results from computational
experiments that investigate the effect of requiring planarity and acyclicity
on the resulting support length. Further, we evaluate the performance and
trade-offs between solution quality and speed of several heuristics relative to
each other and compared to optimal solutions.
