Optimal Sensitivity Oracle for Steiner Mincut

TL;DR

The lower bound is shown 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.

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 $Ω(n^2)$ 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 $Ω(n^2)$ 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.

Figures (4)

• Figure 1: Yellow vertices are Steiner vertices. A mincut for an edge is represented by the same color. $(i)$ Mincuts $A,B$ are for edges $e_1=(s_1,a)$ and $e_2=(s_2,b)$. Observe that $A\cap B$ and $A\cup B$ are Steiner cuts but not mincuts for edges $e_1,e_2$. Moreover, cuts $A\setminus B$ and $B\setminus A$, in which edges $e_1$ and $e_2$ are contributing, are not even Steiner cuts. $(ii)$ Edges $e_1$ and $e_2$ are vital and from Type-3($u$). Mincuts $A$ and $B$ for edges $e_1$ and $e_2$ are crossing, but $A\cap B$ and $A\cup B$ are not Steiner cuts. $(iii)$ Edge $(s,u)$ is from Type-3 and $N((s,u))=\{A,B\}$.
• Figure 2: Illustration of the proof of Lemma \ref{['lem : main lemma for weighted edge']}. $(i)$ There is a Steiner vertex $z$ in $\overline{C_1\cup C_2}$. The red dashed cut shows cut $C_1\cup C_2$ and the blue dashed cut (blue region) shows cut $C_1\cap C_2$. $(ii)$ There is a Steiner vertex $z$ in $C_2\setminus C_1$. Blue dashed cut (blue region) is $C_1\setminus C_2$. Similarly, red dashed cut (light green region) is $C_2\setminus C_1$.
• Figure 3: Graph $G(M)$. Steiner vertices are the yellow vertices and nonSteiner vertices are the blue vertices. Blue edges are of infinite capacities. Every orange edge $(a_i,b_j)$ between set $L$ and $R$ are of capacity $M[i,j]$.
• Figure 4: Transformation of graph $H$ to Graph $G_s(H)$. For mincut $C$ for a vital edge $(u,v)$ in $H$, there is a mincut $C\cup \{s\}$ for vital edge $(u,v)$ in $G_s(H)$.

Theorems & Definitions (43)

• Definition 1: Steiner cut
• Definition 2: single edge Sensitivity Oracle for Steiner mincut
• Theorem 1
• Theorem 2: Sensitivity Oracle for Steiner Mincut
• Theorem 3: Lower Bound for Reporting Capacity
• Theorem 4: Lower Bound for Reporting Cut
• Remark 1
• Definition 3: crossing cuts
• Definition 4: Mincut for an edge
• Definition 5: Vital Edge