The orientation theorem of Nash-Williams states that an undirected graph admits a $k$-arc-connected orientation if and only if it is $2k$-edge-connected. Recently, Ito et al. showed that any orientation of an undirected $2k$-edge-connected graph can be transformed into a $k$-arc-connected orientation by reorienting one arc at a time without decreasing the arc-connectivity at any step, thus providing an algorithmic proof of Nash-Williams' theorem. We generalize their result to hypergraphs and therefore provide an algorithmic proof of the characterization of hypergraphs with a $k$-hyperarc-connected orientation originally given by Frank et al. We prove that any orientation of an undirected $(k,k)$-partition-connected hypergraph can be transformed into a $k$-hyperarc-connected orientation by reorienting one hyperarc at a time without decreasing the hyperarc-connectivity in any step. Furthermore, we provide a simple combinatorial algorithm for computing such a transformation in polynomial time.

翻译：Nash-Williams的定向定理表明：无向图存在$k$-弧连通定向当且仅当其是$2k$-边连通的。近期，Ito等人证明了任意$2k$-边连通无向图的任何定向均可通过逐次重定向单条弧且不降低每一步弧连通性的方式转化为$k$-弧连通定向，从而给出了Nash-Williams定理的算法化证明。本文将他们的结果推广至超图，由此为Frank等人最初提出的具有$k$-超弧连通定向的超图刻画定理提供算法化证明。我们证明：任意$(k,k)$-划分连通无向超图的任何定向可通过逐次重定向单条超弧且不降低每一步超弧连通性的方式转化为$k$-超弧连通定向。此外，我们给出了一个能在多项式时间内计算此类变换的简单组合算法。