License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2021.52
URN: urn:nbn:de:0030-drops-146331
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2021/14633/
He, Zhiyang ;
Li, Jason ;
Wahlström, Magnus
Near-Linear-Time, Optimal Vertex Cut Sparsifiers in Directed Acyclic Graphs
Abstract
Let G be a graph and S, T ⊆ V(G) be (possibly overlapping) sets of terminals, |S| = |T| = k. We are interested in computing a vertex sparsifier for terminal cuts in G, i.e., a graph H on a smallest possible number of vertices, where S ∪ T ⊆ V(H) and such that for every A ⊆ S and B ⊆ T the size of a minimum (A,B)-vertex cut is the same in G as in H. We assume that our graphs are unweighted and that terminals may be part of the min-cut. In previous work, Kratsch and Wahlström (FOCS 2012/JACM 2020) used connections to matroid theory to show that a vertex sparsifier H with O(k³) vertices can be computed in randomized polynomial time, even for arbitrary digraphs G. However, since then, no improvements on the size O(k³) have been shown.
In this paper, we draw inspiration from the renowned Bollobás’s Two-Families Theorem in extremal combinatorics and introduce the use of total orderings into Kratsch and Wahlström’s methods. This new perspective allows us to construct a sparsifier H of Θ(k²) vertices for the case that G is a DAG. We also show how to compute H in time near-linear in the size of G, improving on the previous O(n^{ω+1}). Furthermore, H recovers the closest min-cut in G for every partition (A,B), which was not previously known. Finally, we show that a sparsifier of size Ω(k²) is required, both for DAGs and for undirected edge cuts.
BibTeX - Entry
@InProceedings{he_et_al:LIPIcs.ESA.2021.52,
author = {He, Zhiyang and Li, Jason and Wahlstr\"{o}m, Magnus},
title = {{Near-Linear-Time, Optimal Vertex Cut Sparsifiers in Directed Acyclic Graphs}},
booktitle = {29th Annual European Symposium on Algorithms (ESA 2021)},
pages = {52:1--52:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-204-4},
ISSN = {1868-8969},
year = {2021},
volume = {204},
editor = {Mutzel, Petra and Pagh, Rasmus and Herman, Grzegorz},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/14633},
URN = {urn:nbn:de:0030-drops-146331},
doi = {10.4230/LIPIcs.ESA.2021.52},
annote = {Keywords: graph theory, vertex sparsifier, representative family, matroid}
}
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Keywords: |
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graph theory, vertex sparsifier, representative family, matroid |
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Collection: |
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29th Annual European Symposium on Algorithms (ESA 2021) |
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Issue Date: |
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2021 |
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Date of publication: |
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31.08.2021 |
