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

Authors Mridul Ahi, Keerti Choudhary , Shlok Pande, Pushpraj, Lakshay Saggi

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ACM Subject Classification
  • Mathematics of computing → Paths and connectivity problems
  • Mathematics of computing → Network flows
Keywords
  • Fault tolerance
  • Data structures
  • Minimum cuts
  • Maximum flows

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Abstract

This paper addresses the problem of designing fault-tolerant data structures for the (s,t)-max-flow and (s,t)-min-cut problems in unweighted directed graphs. Given a directed graph 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: 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. This covering property is tight up to constants for single failures and provably cannot extend to comparably small families for k ≥ 2, where we show an Ω(n) lower bound on the family size, independent of λ. 
2) Max-Flow Sensitivity Oracle: Using the fault-tolerant flow family, we construct a single as well as dual-edge sensitivity oracle for (s,t)-max-flow that requires only O(λ 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(n²)-sized oracle for answering (s,t)-min-cut size queries under dual edge failures in constant time, along with a matching lower bound. We extend this by focusing on graphs with small min-cut values λ, and present a more compact oracle of size O(λ n) that answers such min-cut size queries in constant time and reports the corresponding (s,t)-min-cut partition in O(n) time. We also show that the space complexity of our oracle is asymptotically optimal in this setting. 
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. This also leads to improved fault-tolerant (s,t)-reachability oracles, achieving O(n log n) space and O(log n) query time for up to k = O(1) edge failures.

Cite As Get BibTex

Mridul Ahi, Keerti Choudhary, Shlok Pande, Pushpraj, and Lakshay Saggi. Maximum-Flow and Minimum-Cut Sensitivity Oracles for Directed Graphs. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 5:1-5:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.ITCS.2026.5

Author Details

Mridul Ahi
  • Department of Mathematics, IIT Delhi, India
Keerti Choudhary
  • Department of Computer Science and Engineering, IIT Delhi, India
Shlok Pande
  • Department of Computer Science and Engineering, IIT Delhi, India
Pushpraj
  • Department of Computer Science and Engineering, IIT Delhi, India
Lakshay Saggi
  • Department of Computer Science and Engineering, IIT Delhi, India

Funding

  • Choudhary, Keerti: Partially supported by Google India Research Awards.
  • Saggi, Lakshay: Supported by Prime Minister's Research Fellowship (PMRF).

Acknowledgements

The authors would like to thank Thatchaphol Saranurak, Surender Baswana, and Koustav Bhanja for helpful discussions.

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