Minimum+1 Steiner Cuts and Dual Edge Sensitivity Oracle: Bridging the Gap between Global cut and (s,t)-cut

TL;DR

This work studies Steiner cuts of capacity minimum+1, and as an important application, it provides a dual edge Sensitivity Oracle for Steiner mincut, and presents the following first results.

Abstract

Let $G=(V,E)$ be an undirected multi-graph on $n=|V|$ vertices and $S\subseteq V$ be a Steiner set. Steiner cut is a fundamental concept; moreover, global cut $(|S|=n)$, as well as (s,t)-cut $(|S|=2)$, is just a special case of Steiner cut. We study Steiner cuts of capacity minimum+1, and as an important application, we provide a dual edge Sensitivity Oracle for Steiner mincut. A compact data structure for cuts of capacity minimum+1 has been designed for both global cuts [STOC 1995] and (s,t)-cuts [TALG 2023]. Moreover, both data structures are also used crucially to design a dual edge Sensitivity Oracle for their respective mincuts. Unfortunately, except for these two extreme scenarios of Steiner cuts, no generalization of these results is known. Therefore, to address this gap, we present the following first results on Steiner cuts. 1. Data Structure: There is an $O(n(n-|S|+1))$ space data structure that can determine in $O(1)$ time whether a given pair of vertices is separated by a Steiner cut of capacity at least minimum+1. It can report such a cut, if it exists, in $O(n)$ time. 2. Sensitivity Oracle: (a) There is an $O(n(n-|S|+1))$ space data structure that, after the failure/insertion of any pair of edges, can report the capacity of Steiner mincut in $O(1)$ time and a Steiner mincut in $O(n)$ time. (b) If we are interested in reporting only the capacity, there is a more compact data structure that occupies $O((n-|S|)^2+n)$ space and reports the capacity in $O(1)$ time after the failure/insertion of any pair of edges. 3. Lower Bound: For undirected multi-graphs, for every Steiner set $S$, any data structure that, after the failure or insertion of any pair of edges, can report the capacity of Steiner mincut must occupy $Ω((n-|S|)^2)$ bits of space, irrespective of the query time.

Figures (7)

• Figure 1: The vertices $\{s,t_1,t_2,\ldots t_q\}$ are Steiner vertices of $G({\mathcal{W}})$ and $N_{S({\mathcal{W}})}(u)=\{C_1,C_2,\ldots, C_q\}$. $(i)$ Let $q=\Omega(|{\mathcal{W}}|)$. The capacity of global mincut of this graph is at least $4$ for $q\ge 2$. $C$ and $C'$ are Steiner cuts of capacity $2q+1$ and $2q$ respectively. The capacity of S-mincut is $2q$. Each cut $C_i$, $1\le i\le q$, has capacity $2q+1$; and moreover, $C_i$ has $t_2$ in $\overline{C_i}$. $(ii)$ Let $q=|S({\mathcal{W}})|$. For every pair of cuts $C_i,C_j$, $1\le i\ne j\le q$, from $N_{S({\mathcal{W}})}(u)$, a vertex $v$ from ${\mathcal{W}}$ belongs to $\overline{C_i\cup C_j}$. $(iii)$ Construction of $G({\mathcal{W}})$ from $G'$. The same color vertices not belonging to ${\mathcal{W}}$ are contracted into a Steiner vertex in $G({\mathcal{W}})$.
• Figure 2: Depicting all the eight regions formed by three $(\lambda_S+1)$ cuts $C_1,C_2,$ and $C_3$ of $G$.
• Figure 3: $(i)$ The set $\overline{C_1\cup C_2\cup C_3}$ of any three cuts $C_1,C_2,C_3\in N_{S({\mathcal{W}})}(u)$ is empty, shown in red region. $(ii)$ Data structure ${\mathcal{N}}_{{\mathcal{W}}}^{\ge 4}(u)$ for ${\mathcal{W}}$. Vertex $y$ cannot belong to more than two arrays.
• Figure 4: (i) Illustration of the proof of Lemma \ref{['lem : property p3']} and Lemma \ref{['lem : necessary condition']}. Green and Yellow curves represent the Steiner cuts $C\setminus C'$ and $C'\setminus C$, respectively. $(ii)$ Illustration of the proof of Lemma \ref{['lem : sufficient condition']}.
• Figure 5: Yellow vertices are Steiner vertices of $G({\mathcal{W}})$ and white vertices belong to ${\mathcal{W}}$. Let $q={\Omega}(|S({\mathcal{W}})|)$. In this graph, the capacity of S-mincut is $4$, $C'\in N_{S({\mathcal{W}})}(u_1)$ and $C\in N_{S({\mathcal{W}})}(u_2)$. There are $q$ Steiner vertices that belong to $\overline{C_1\cup C_2}$.

Theorems & Definitions (143)

• Definition 1: Steiner cut
• Definition 2: Dual edge Sensitivity Oracle
• Theorem 1: Data Structure
• Theorem 2: Sensitivity Oracle
• Theorem 3: Lower Bound
• Lemma 1: Sub-modularity of Cuts DBLP:journals/dam/NagamochiI00
• Lemma 2
• Definition 3
• Definition 4: Crossing cuts and Corner sets
• Definition 5: Nearest $(\lambda_S+1)$ cut of a vertex