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.

翻译：我们研究超图网络设计问题，其目标是在满足特定连通性要求的前提下构建超图。对于图网络设计问题而言，每个可行解中的边数至多为顶点数的二次方。相比之下，超图网络设计问题的可行解中可能包含指数级于顶点数的超边数量。这使得超图网络设计问题相较于图网络设计问题面临额外技术挑战：为在多项式时间内求解，首先需证明存在可行解，其超边数量与输入规模呈多项式关系。本工作的核心在于证明特定超图网络设计问题存在解，其超边数量与顶点数呈多项式关系，且可通过强多项式时间算法求解。我们改进了这些问题此前最快的伪多项式运行时间。此外，我们还开发了强多项式时间算法，能返回近均匀超图作为解（即任意两条超边的大小之差至多为1）。作为研究成果的应用，我们首次推导出以下问题的强多项式时间算法：(i) 使用超边的度规格化超图连通性增强；(ii) 使用超边的度规格化超图节点-区域连通性增强；(iii) 使用超边的度约束混合超图连通性增强。