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When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ICALP.2022.15
URN: urn:nbn:de:0030-drops-163566
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16356/
Baswana, Surender ;
Bhanja, Koustav ;
Pandey, Abhyuday
Minimum+1 (s,t)-cuts and Dual Edge Sensitivity Oracle
Abstract
Let G be a directed multi-graph 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):8-16, 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 509-518, 1995] showed that there exists an ?(n) size 2-level 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 3-transversal - 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 ?(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 ?(|B|) time.
3) There exists an O(n²) size data structure that solves the dual edge sensitivity problem for (s,t)-mincuts. It takes ?(1) time to report the value of a resulting (s,t)-mincut (A,B) and ?(|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 – all-pairs 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 = {15:1--15:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-235-8},
ISSN = {1868-8969},
year = {2022},
volume = {229},
editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16356},
URN = {urn:nbn:de:0030-drops-163566},
doi = {10.4230/LIPIcs.ICALP.2022.15},
annote = {Keywords: mincut, maxflow, fault tolerant}
}
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Keywords: |
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mincut, maxflow, fault tolerant |
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Collection: |
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49th International Colloquium on Automata, Languages, and Programming (ICALP 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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28.06.2022 |
