We consider hypergraph network design problems where the goal is to construct a hypergraph satisfying certain properties. In graph network design problems, the number of edges in an arbitrary solution is at most the square of the number of vertices. In contrast, in hypergraph network design problems, the number of hyperedges in an arbitrary solution could be exponential in the number of vertices and hence, additional care is necessary to design polynomial-time algorithms. The central theme of this work is to show that certain hypergraph network design problems admit solutions with polynomial number of hyperedges 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 algorithms that return (near-)uniform hypergraphs as solutions. The hypergraph network design problems that we focus upon are splitting-off operation in hypergraphs, connectivity augmentation using hyperedges, and covering skew-supermodular functions using hyperedges. Our definition of the splitting-off operation in hypergraphs and our proof showing the existence of the operation using a strongly polynomial-time algorithm to compute it are likely to be of independent graph-theoretical interest.

翻译：我们考虑超图网络设计问题，目标在于构造满足特定性质的超图。在图网络设计问题中，任意解中的边数至多是顶点数的平方。相比之下，在超图网络设计问题中，任意解中的超边数可能随顶点数呈指数增长，因此设计多项式时间算法需要格外谨慎。本工作的核心主题是证明某些超图网络设计问题存在超边数为多项式的解，且可在强多项式时间内求解。我们的工作改进了这些问题的此前最快伪多项式运行时间。此外，我们开发的算法能返回（近似）均匀超图作为解。本文重点关注的超图网络设计问题包括：超图中的分裂操作、使用超边进行连通性增强、以及使用超边覆盖斜超模函数。我们对超图中分裂操作的定义，以及通过强多项式时间算法证明该操作存在性的过程，可能具有独立的图论研究价值。