Narayanan showed the existence of the principal partition sequence of a submodular function, a structure with numerous applications in areas such as clustering, fast algorithms, and approximation algorithms. In this work, motivated by two applications, we develop a theory of $\{s,t\}$-separating principal partition sequence of a submodular function. We define this sequence, show its existence, and design a polynomial-time algorithm to construct it. We show two applications: (1) approximation algorithm for the $\{s,t\}$-separating submodular $k$-partitioning problem for monotone and posimodular functions and (2) polynomial-time algorithm for the hypergraph orientation problem of finding an orientation that simultaneously has strong connectivity at least $k$ and $(s,t)$-connectivity at least $\ell$.

翻译：Narayanan 证明了子模函数主划分序列的存在性，该结构在聚类、快速算法和近似算法等领域具有广泛应用。在本工作中，受两个应用问题的驱动，我们发展了子模函数的{s,t}-分离主划分序列理论。我们定义了该序列，证明了其存在性，并设计了一个多项式时间算法来构造它。我们展示了两个应用：(1) 针对单调且正模函数的{s,t}-分离子模k-划分问题的近似算法；(2) 超图定向问题的多项式时间算法，该问题旨在找到一个同时具有至少k的强连通性和至少ℓ的(s,t)-连通性的定向。