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.APPROX/RANDOM.2022.32
URN: urn:nbn:de:0030-drops-171544
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/17154/
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Apers, Simon ; Gawrychowski, Paweł ; Lee, Troy

Finding the KT Partition of a Weighted Graph in Near-Linear Time

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Abstract

In a breakthrough work, Kawarabayashi and Thorup (J. ACM'19) gave a near-linear time deterministic algorithm to compute the weight of a minimum cut in a simple graph G = (V,E). A key component of this algorithm is finding the (1+ε)-KT partition of G, the coarsest partition {P_1, …, P_k} of V such that for every non-trivial (1+ε)-near minimum cut with sides {S, ̄{S}} it holds that P_i is contained in either S or ̄{S}, for i = 1, …, k. In this work we give a near-linear time randomized algorithm to find the (1+ε)-KT partition of a weighted graph. Our algorithm is quite different from that of Kawarabayashi and Thorup and builds on Karger’s framework of tree-respecting cuts (J. ACM'00).
We describe a number of applications of the algorithm. (i) The algorithm makes progress towards a more efficient algorithm for constructing the polygon representation of the set of near-minimum cuts in a graph. This is a generalization of the cactus representation, and was initially described by Benczúr (FOCS'95). (ii) We improve the time complexity of a recent quantum algorithm for minimum cut in a simple graph in the adjacency list model from Õ(n^{3/2}) to Õ(√{mn}), when the graph has n vertices and m edges. (iii) We describe a new type of randomized algorithm for minimum cut in simple graphs with complexity ?(m + n log⁶ n). For graphs that are not too sparse, this matches the complexity of the current best ?(m + n log² n) algorithm which uses a different approach based on random contractions.
The key technical contribution of our work is the following. Given a weighted graph G with m edges and a spanning tree T of G, consider the graph H whose nodes are the edges of T, and where there is an edge between two nodes of H iff the corresponding 2-respecting cut of T is a non-trivial near-minimum cut of G. We give a ?(m log⁴ n) time deterministic algorithm to compute a spanning forest of H.

BibTeX - Entry

@InProceedings{apers_et_al:LIPIcs.APPROX/RANDOM.2022.32,
  author =	{Apers, Simon and Gawrychowski, Pawe{\l} and Lee, Troy},
  title =	{{Finding the KT Partition of a Weighted Graph in Near-Linear Time}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)},
  pages =	{32:1--32:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-249-5},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{245},
  editor =	{Chakrabarti, Amit and Swamy, Chaitanya},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2022/17154},
  URN =		{urn:nbn:de:0030-drops-171544},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2022.32},
  annote =	{Keywords: Graph theory}
}

Keywords: Graph theory
Collection: Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)
Issue Date: 2022
Date of publication: 15.09.2022

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