## 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 naïve 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 δ ∈ (0, 1] there is an O(m · n^{δ})-time algorithm that provides, w.h.p., cycles of length O(log n)^{1/δ} 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}/ with n^{1+} · 2^{1/} extra edges, for every ∈ (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/δ} edges for some constant δ. These decomposition algorithms lead to improvements in all the algorithmic applications of Chu et al. as well as to new distributed constructions.

Original language | English |
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Title of host publication | 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 |

Editors | Christel Baier, Ioannis Chatzigiannakis, Paola Flocchini, Stefano Leonardi |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

ISBN (Electronic) | 9783959771092 |

DOIs | |

State | Published - 1 Jul 2019 |

Externally published | Yes |

Event | 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 - Patras, Greece Duration: 9 Jul 2019 → 12 Jul 2019 |

### Publication series

Name | Leibniz International Proceedings in Informatics, LIPIcs |
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Volume | 132 |

ISSN (Print) | 1868-8969 |

### Conference

Conference | 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019 |
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Country/Territory | Greece |

City | Patras |

Period | 9/07/19 → 12/07/19 |

### Bibliographical note

Publisher Copyright:© Merav Parter and Eylon Yogev; licensed under Creative Commons License CC-BY

### Funding

Funding Merav Parter: The Israel Science Foundation grant no. 2084/18 Eylon Yogev: The European Union’s Horizon 2020 research and innovation program under grant agreement No. 742754. Merav Parter: The Israel Science Foundation grant no. 2084/18 Eylon Yogev: The European Union's Horizon 2020 research and innovation program under grant agreement No. 742754.

Funders | Funder number |
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Horizon 2020 Framework Programme | 742754 |

Israel Science Foundation | 2084/18 |

Horizon 2020 |

## Keywords

- Cycle decomposition
- Graph sparsification
- Low-congestion cycle cover