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

2025-11-28

AI Summary

This paper addresses sensitivity queries for maximum flow and minimum cut in directed graphs under edge failures. Method: We introduce the first compact multi-failure sensitivity oracle, built upon fault-tolerant flow families and (s,t)-maximum flow decomposition, augmented with dynamic graph projection and hierarchical tree encoding. Contribution/Results: For two failures, our oracle occupies only O(λn) space—where λ is the s–t maximum flow value—and supports O(1)-time queries for flow value or cut size, and O(n)-time reconstruction of the cut set. Extending to k failures, it achieves O_{λ,k}(n log n) space and O_{λ,k}(log n) query time. Notably, this is the first solution achieving linear space and logarithmic query time for minimum cut sensitivity under multiple failures in graphs with small cut values, markedly improving over prior exponential-space constructions.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Constructs fault-tolerant flow families for directed graphs with edge failures.
• Develops compact oracles for max-flow sensitivity under single or dual edge failures.
• Extends min-cut sensitivity oracles to handle multiple edge failures efficiently.

Innovation

• Constructs fault-tolerant flow family for edge failures
• Develops compact max-flow sensitivity oracle with O(λn) space
• Extends min-cut oracle for multiple failures with efficient queries