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DOI: 10.4230/LIPIcs.ICALP.2019.89
URN: urn:nbn:de:0030-drops-106653
URL: http://drops.dagstuhl.de/opus/volltexte/2019/10665/
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Parter, Merav ; Yogev, Eylon

Optimal Short Cycle Decomposition in Almost Linear Time

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Abstract

Short cycle decomposition is an edge partitioning of an unweighted graph into edge-disjoint short cycles, plus a small number of extra edges not in any cycle. This notion was introduced by Chu et al. [FOCS'18] as a fundamental tool for graph sparsification and sketching. Clearly, it is most desirable to have a fast algorithm for partitioning the edges into as short as possible cycles, while omitting few edges. The most na´ve procedure for such decomposition runs in time O(m * n) and partitions the edges into O(log n)-length edge-disjoint cycles plus at most 2n edges. Chu et al. improved the running time considerably to m^{1+o(1)}, while increasing both the length of the cycles and the number of omitted edges by a factor of n^{o(1)}. Even more recently, Liu-Sachdeva-Yu [SODA'19] showed that for every constant delta in (0,1] there is an O(m * n^{delta})-time algorithm that provides, w.h.p., cycles of length O(log n)^{1/delta} and O(n) extra edges. In this paper, we significantly improve upon these bounds. We first show an m^{1+o(1)}-time deterministic algorithm for computing nearly optimal cycle decomposition, i.e., with cycle length O(log^2 n) and an extra subset of O(n log n) edges not in any cycle. This algorithm is based on a reduction to low-congestion cycle covers, introduced by the authors in [SODA'19]. We also provide a simple deterministic algorithm that computes edge-disjoint cycles of length 2^{1/epsilon} with n^{1+epsilon}* 2^{1/epsilon} extra edges, for every epsilon in (0,1]. Combining this with Liu-Sachdeva-Yu [SODA'19] gives a linear time randomized algorithm for computing cycles of length poly(log n) and O(n) extra edges, for every n-vertex graphs with n^{1+1/delta} edges for some constant delta. These decomposition algorithms lead to improvements in all the algorithmic applications of Chu et al. as well as to new distributed constructions.

BibTeX - Entry

@InProceedings{parter_et_al:LIPIcs:2019:10665,
  author =	{Merav Parter and Eylon Yogev},
  title =	{{Optimal Short Cycle Decomposition in Almost Linear Time}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{89:1--89:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-109-2},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{132},
  editor =	{Christel Baier and Ioannis Chatzigiannakis and Paola Flocchini and Stefano Leonardi},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2019/10665},
  URN =		{urn:nbn:de:0030-drops-106653},
  doi =		{10.4230/LIPIcs.ICALP.2019.89},
  annote =	{Keywords: Cycle decomposition, low-congestion cycle cover, graph sparsification}
}

Keywords: Cycle decomposition, low-congestion cycle cover, graph sparsification
Seminar: 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)
Issue Date: 2019
Date of publication: 08.07.2019


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