Let $G=(V,E)$ be an undirected weighted graph on $n=|V|$ vertices and $S\subseteq V$ be a Steiner set. Steiner mincut is a well-studied concept, which provides a generalization to both (s,t)-mincut (when $|S|=2$) and global mincut (when $|S|=n$). Here, we address the problem of designing a compact data structure that can efficiently report a Steiner mincut and its capacity after the failure of any edge in $G$; such a data structure is known as a \textit{Sensitivity Oracle} for Steiner mincut. In the area of minimum cuts, although many Sensitivity Oracles have been designed in unweighted graphs, however, in weighted graphs, Sensitivity Oracles exist only for (s,t)-mincut [Annals of Operations Research 1991, NETWORKS 2019, ICALP 2024], which is just a special case of Steiner mincut. Here, we generalize this result to any arbitrary set $S\subseteq V$. 1. Sensitivity Oracle: Assuming the capacity of every edge is known, a. there is an ${\mathcal O}(n)$ space data structure that can report the capacity of Steiner mincut in ${\mathcal O}(1)$ time and b. there is an ${\mathcal O}(n(n-|S|+1))$ space data structure that can report a Steiner mincut in ${\mathcal O}(n)$ time after the failure of any edge in $G$. 2. Lower Bound: We show that any data structure that, after the failure of any edge, can report a Steiner mincut or its capacity must occupy $\Omega(n^2)$ bits of space in the worst case, irrespective of the size of the Steiner set. The lower bound in (2) shows that the assumption in (1) is essential to break the $\Omega(n^2)$ lower bound on space. For $|S|=n-k$ for any constant $k\ge 0$, it occupies only ${\mathcal O}(n)$ space. So, we also present the first Sensitivity Oracle occupying ${\mathcal O}(n)$ space for global mincut.

翻译：令$G=(V,E)$为一个具有$n=|V|$个顶点的无向加权图，$S\subseteq V$为一个Steiner点集。Steiner最小割是一个被深入研究的经典概念，它同时推广了(s,t)-最小割（当$|S|=2$时）和全局最小割（当$|S|=n$时）。本文研究如何设计一个紧凑的数据结构，使其能够在$G$中任意边失效后，高效地报告Steiner最小割及其容量；此类数据结构被称为Steiner最小割的\textit{敏感性预言机}。在最小割研究领域中，尽管已有许多针对无权图的敏感性预言机被设计出来，但在加权图中，现有的敏感性预言机仅针对(s,t)-最小割[Annals of Operations Research 1991, NETWORKS 2019, ICALP 2024]，而这只是Steiner最小割的一个特例。本文将此结果推广至任意点集$S\subseteq V$。1. 敏感性预言机：在已知每条边容量的前提下，a. 存在一个占用${\mathcal O}(n)$空间的数据结构，可在${\mathcal O}(1)$时间内报告Steiner最小割的容量；b. 存在一个占用${\mathcal O}(n(n-|S|+1))$空间的数据结构，可在$G$中任意边失效后，于${\mathcal O}(n)$时间内报告一个Steiner最小割。2. 下界：我们证明，任何能够在任意边失效后报告Steiner最小割或其容量的数据结构，在最坏情况下必须占用$\Omega(n^2)$比特的存储空间，且该下界与Steiner点集的大小无关。下界(2)表明，结果(1)中的前提假设对于突破$\Omega(n^2)$的空间下界是至关重要的。对于任意常数$k\ge 0$，当$|S|=n-k$时，该数据结构仅占用${\mathcal O}(n)$空间。因此，我们也首次提出了一个仅占用${\mathcal O}(n)$空间的全局最小割敏感性预言机。