License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
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DOI: 10.4230/LIPIcs.ESA.2021.52
URN: urn:nbn:de:0030-drops-146331
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He, Zhiyang ; Li, Jason ; Wahlström, Magnus

Near-Linear-Time, Optimal Vertex Cut Sparsifiers in Directed Acyclic Graphs

LIPIcs-ESA-2021-52.pdf (0.8 MB)


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

  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 =		{},
  URN =		{urn:nbn:de:0030-drops-146331},
  doi =		{10.4230/LIPIcs.ESA.2021.52},
  annote =	{Keywords: graph theory, vertex sparsifier, representative family, matroid}

Keywords: graph theory, vertex sparsifier, representative family, matroid
Collection: 29th Annual European Symposium on Algorithms (ESA 2021)
Issue Date: 2021
Date of publication: 31.08.2021

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