  We consider hypergraph network design problems where the goal is to construct
a hypergraph that satisfies certain connectivity requirements. For graph
network design problems where the goal is to construct a graph that satisfies
certain connectivity requirements, the number of edges in every feasible
solution is at most quadratic in the number of vertices. In contrast, for
hypergraph network design problems, we might have feasible solutions in which
the number of hyperedges is exponential in the number of vertices. This
presents an additional technical challenge in hypergraph network design
problems compared to graph network design problems: in order to solve the
problem in polynomial time, we first need to show that there exists a feasible
solution in which the number of hyperedges is polynomial in the input size.
  The central theme of this work is to show that certain hypergraph network
design problems admit solutions in which the number of hyperedges is polynomial
in the number of vertices and moreover, can be solved in strongly polynomial
time. Our work improves on the previous fastest pseudo-polynomial run-time for
these problems. In addition, we develop strongly polynomial time algorithms
that return near-uniform hypergraphs as solutions (i.e., every pair of
hyperedges differ in size by at most one). As applications of our results, we
derive the first strongly polynomial time algorithms for (i) degree-specified
hypergraph connectivity augmentation using hyperedges, (ii) degree-specified
hypergraph node-to-area connectivity augmentation using hyperedges, and (iii)
degree-constrained mixed-hypergraph connectivity augmentation using hyperedges.
