Abstract
Let G be a directed multigraph on n vertices and m edges with a designated source vertex s and a designated sink vertex t. We study the (s,t)cuts of capacity minimum+1 and as an important application of them, we give a solution to the dual edge sensitivity for (s,t)mincuts  reporting the (s,t)mincut upon failure or addition of any pair of edges.
Picard and Queyranne [Mathematical Programming Studies, 13(1):816, 1980] showed that there exists a directed acyclic graph (DAG) that compactly stores all minimum (s,t)cuts of G. This structure also acts as an oracle for the single edge sensitivity of minimum (s,t)cut. Dinitz and Nutov [STOC, pages 509518, 1995] showed that there exists an 𝒪(n) size 2level cactus model that stores all global cuts of capacity minimum+1. However, for minimum+1 (s,t)cuts, no such compact structures exist till date. We present the following structural and algorithmic results on minimum+1 (s,t)cuts.
1) There exists a pair of DAGs of size O(m) that compactly store all minimum+1 (s,t)cuts of G. Each minimum+1 (s,t)cut appears as a (s,t)cut in one of the 2 DAGs and is 3transversal  it intersects any path in the DAG at most thrice.
2) There exists an O(n²) size data structure that, given a pair of vertices {u,v} which are not separated by an (s,t)mincut, can determine in 𝒪(1) time if there exists a minimum+1 (s,t)cut, say (A,B), such that {s,u} ∈ A and {v,t} ∈ B; the corresponding cut can be reported in 𝒪(B) time.
3) There exists an O(n²) size data structure that solves the dual edge sensitivity problem for (s,t)mincuts. It takes 𝒪(1) time to report the value of a resulting (s,t)mincut (A,B) and 𝒪(B) time to report the cut.
4) For the data structure problems addressed in (2) and (3) above, we also provide a matching conditional lower bound. We establish a close relationship among three seemingly unrelated problems – allpairs directed reachability problem, the dual edge sensitivity problem for (s,t)mincuts, and 2× 2 maximum flow. Assuming the directed reachability hypothesis, this leads to Ω(n²) lower bounds on the space for the latter two problems.
BibTeX  Entry
@InProceedings{baswana_et_al:LIPIcs.ICALP.2022.15,
author = {Baswana, Surender and Bhanja, Koustav and Pandey, Abhyuday},
title = {{Minimum+1 (s,t)cuts and Dual Edge Sensitivity Oracle}},
booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)},
pages = {15:115:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772358},
ISSN = {18688969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16356},
URN = {urn:nbn:de:0030drops163566},
doi = {10.4230/LIPIcs.ICALP.2022.15},
annote = {Keywords: mincut, maxflow, fault tolerant}
}
Keywords: 

mincut, maxflow, fault tolerant 
Collection: 

49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) 
Issue Date: 

2022 
Date of publication: 

28.06.2022 