{"ID":2835657,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.00153","arxiv_id":"2512.00153","title":"Maximum-Flow and Minimum-Cut Sensitivity Oracles for Directed Graphs","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.","short_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. Fau...","url_abs":"https://arxiv.org/abs/2512.00153","url_pdf":"https://arxiv.org/pdf/2512.00153v1","authors":"[\"Mridul Ahi\",\"Keerti Choudhary\",\"Shlok Pande\",\"Pushpraj\",\"Lakshay Saggi\"]","published":"2025-11-28T19:00:01Z","proceeding":"cs.DS","tasks":"[\"cs.DS\"]","methods":"[]","has_code":false}