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

Given a digraph $G = (V, E)$ with a designated source $s$, sink $t$, and an $(s,t)$-max-flow of value $\lambda$, 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 $\lambda$, we construct a family $B$ of $2\lambda+1$ $(s,t)$-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 $O(\lambda n)$ 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 $\lambda$, and present a more compact oracle of size $O(\lambda n)$ 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 $\lambda$, we construct a $k$-fault-tolerant min-cut oracle with space complexity $O_{\lambda,k}(n \log n)$ that answers min-cut size queries in $O_{\lambda,k}(\log n)$ time.