Vital Edges for (s,t)-mincut: Efficient Algorithms, Compact Structures, and Optimal Sensitivity Oracle

TL;DR

The first and optimal SO for DiGraph, a directed weighted graph on n vertices and m edges with source s and sink t, is presented, which solves an open problem stated by Ausiello et al.

Abstract

Let G be a directed weighted graph (DiGraph) on n vertices and m edges with source s and sink t. An edge in G is vital if its removal reduces the capacity of (s,t)-mincut. Since the seminal work of Ford and Fulkerson, a long line of work has been done on computing the most vital edge and all vital edges of G. Unfortunately, after 60 years, the existing results are for undirected or unweighted graphs. We present the following result for DiGraph, which solves an open problem stated by Ausiello et al. 1. There is an algorithm that computes all vital edges as well as the most vital edge of G using O(n) maxflow computations. Vital edges play a crucial role in the design of Sensitivity Oracle (SO) for (s,t)-mincut. For directed graphs, the only existing SO is for unweighted graphs by Picard and Queyranne. We present the first and optimal SO for DiGraph. 2. (a) There is an O(n) space SO that can report in O(1) time the capacity of (s,t)-mincut and (b) an O($n^2$) space SO that can report an (s,t)-mincut in O(n) time after failure/insertion of an edge. For unweighted graphs, Picard and Queyranne designed an O(m) space DAG that stores and characterizes all mincuts for all vital edges. Conversely, there is a set containing at most n-1 (s,t)-cuts such that at least one mincut for every vital edge belongs to the set. We generalize these results for DiGraph. 3. (a) There is a set containing at most n-1 (s,t)-cuts such that at least one mincut for every vital edge is present in the set. (b) We design two compact structures for storing and characterizing all mincuts for all vital edges, (i) O(m) space DAG for partial characterization and (ii) O(mn) space structure for complete characterization. To arrive at our results, we develop new techniques, especially a generalization of maxflow-mincut theorem by Ford and Fulkerson, which might be of independent interest.

Figures (10)

• Figure 1: Each mincut for vital edge $(v_1,v_4)$ contains a nonvital edge (shown in the same color).
• Figure 2: A mincut cover fails to include mincut $B$ for edges $(a,t)$ and $(s,b)$ if property ${\mathcal{P}}$ is not ensured in the construction. An edge and the mincut for the edge are shown in the same color.
• Figure 3: $(i)$ A graph $H$ and $(ii)$${\mathcal{D}}_{PQ}(H)$. Thick edges in $(i)$ represent the vital edges of $H$ that are internal to the nodes of ${\mathcal{D}}_{PQ}(H)$. A mincut for them is represented by dashed curves.
• Figure 4: $A$ is the $(s,t)$-cut of least capacity that separates $a$ and $b$. Edge $(a,b)$ is a vital edge, and it is incoming to $A$.
• Figure 5: ($i$) $\Gamma$-edges between a pair of mincuts $C_1$ and $C_2$ for a pair of vital edges. ($ii$) mincuts $A$ and $B$ for vital edges $(c,t)$ and $(d,t)$, respectively, are not closed under union. Moreover, edge $(b,a)$ (likewise edge $(c,d)$) contributes to $A$ (likewise to $B$) and is incoming to $B$ (likewise to $A$).

Theorems & Definitions (88)

• Definition 1: vital edge
• Definition 2: mincut for an edge
• Theorem 1.1: Mincut Cover
• Theorem 1.2: Sensitivity Oracle
• Theorem 1.3: Sensitivity Oracle for Reporting Capacity
• Theorem 1.4
• Remark 1
• Theorem 1.5: Computing All Vital Edges
• Theorem 1.6
• Theorem 1.7: Partial Characterization