Maximum-Flow and Minimum-Cut Sensitivity Oracles for Directed Graphs

TL;DR

<3-5 sentence high-level summary> This work develops compact, fault-tolerant data-structures (oracles) for directed graphs to support max-flow and min-cut queries under edge failures. It introduces a fault-tolerant flow family of size 2λ+1, enabling efficient single- and dual-edge max-flow updates with space O(λn) and fast query times, and it derives dual-failure min-cut oracles with matching space bounds, plus a k-fault-tolerant min-cut oracle via randomized augmentation. The methods leverage circulation with lower bounds, SCC preservers, and specialized graph augmentations to convert minimal cuts into min-cuts, achieving near-linear space for small failure budgets and providing both structural insights and practical update mechanisms. These results advance fault-tolerant network analysis by enabling rapid, scalable updates to fundamental flow and cut quantities in directed graphs under small numbers of failures.

Abstract

Given a digraph $G = (V, E)$ with a designated source $s$, sink $t$, and an $(s,t)$-max-flow of value $λ$, we present constructions for max-flow and min-cut sensitivity oracles, and introduce the concept of a fault-tolerant flow family, which may be of independent interest. Our main contributions are as follows. 1. Fault-Tolerant Flow Family: For any graph $G$ with $(s,t)$-max-flow value $λ$, we construct a family $B$ of $2λ+1$ $(s,t)$-flows such that for every edge $e$, $B$ contains an $(s,t)$-max-flow of $G-e$. 2. Max-Flow Sensitivity Oracle: We construct a single as well as dual-edge sensitivity oracle for $(s,t)$-max-flow that requires only $O(λn)$ space. Given any set $F$ of up to two failing edges, the oracle reports the updated max-flow value in $G-F$ in $O(n)$ time. Additionally, for the single-failure case, the oracle can determine in constant time whether the flow through an edge $x$ changes when another edge $e$ fails. 3. Min-Cut Sensitivity Oracle for Dual Failures: Recently, Baswana et al. (ICALP'22) designed an $O(n^2)$-sized oracle for answering $(s,t)$-min-cut size queries under dual edge failures in constant time. We extend this by focusing on graphs with small min-cut values $λ$, and present a more compact oracle of size $O(λn)$ that answers such min-cut size queries in constant time and reports the corresponding $(s,t)$-min-cut partition in $O(n)$ time. 4. Min-Cut Sensitivity Oracle for Multiple Failures: We extend our results to the general case of $k$ edge failures. For any graph with $(s,t)$-min-cut of size $λ$, we construct a $k$-fault-tolerant min-cut oracle with space complexity $O_{λ,k}(n \log n)$ that answers min-cut size queries in $O_{λ,k}(\log n)$ time.

Figures (6)

• Figure 1: A pair of two $(s,t)$ paths intersecting at $(s,t)$-cuts of size one.
• Figure 2: The SCCs in the residual graph $G_f$ may split on deletion of $e$, which makes some intra-SCC edges as inter-SCC in the updated residual graph.
• Figure 3: Depiction of graphs $G_\lambda$ and $D_\lambda$. Note that the strip graph $D_\lambda$ is precisely the DAG of the SCCs of $G_f$, but with all edge directions reversed.
• Figure 4: Lower bound construction for dual fault-tolerant min-cut oracle.
• Figure 5: Depiction of nearest $(\min+1)$-cuts in $G$ such that $x_i,u_W$ lie on the source side.

Theorems & Definitions (44)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Corollary 1.6
• Definition 2.1
• Definition 2.2: $(Min+k)$-cut
• Definition 2.3: Critical edge
• Lemma 2.4