Optimal Sensitivity Oracle for Steiner Mincut

TL;DR

This paper introduces a new sensitivity oracle for Steiner mincut in weighted graphs, enabling efficient updates after edge failures, with proven lower bounds on space complexity and optimal solutions for special cases.

Contribution

It generalizes sensitivity oracles to weighted graphs for any Steiner set, providing space-efficient data structures and establishing fundamental lower bounds.

Findings

O(n) space oracle reports Steiner mincut capacity in O(1) time

O(n(n-|S|+1)) space oracle reports Steiner mincut in O(n) time

Lower bound of Ω(n^2) bits on space for general Steiner mincut sensitivity oracles

Abstract

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.