## Abstract

We present several new effcient algorithms for approximating the girth, g, of weighted and unweighted n-vertex, m-edge undirected graphs. For undirected graphs with polynomially bounded, integer, non-negative edge weights, we provide an algorithm that for every integer k 1, runs in eO(m + n1+1=k log g) time and returns a cycle of length at most 2kg. For unweighted, undirected graphs we present an algorithm that for every k 1, runs in eO (n1+1=k) time and returns a cycle of length at most 2kdg=2e, an almost k-approximation. Both algorithms provide trade-off-s between the running time and the quality of the approximation. We also obtain faster algorithms for approximation factors better than 2, and improved approximations when the girth is odd or small (e.g., 3 and 4).

Original language | English |
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Title of host publication | ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 |

Publisher | Association for Computing Machinery |

Pages | 1471-1492 |

Number of pages | 22 |

ISBN (Electronic) | 9781611977073 |

State | Published - 2022 |

Event | 33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 - Alexander, United States Duration: 9 Jan 2022 → 12 Jan 2022 |

### Publication series

Name | Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms |
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Volume | 2022-January |

### Conference

Conference | 33rd Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2022 |
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Country/Territory | United States |

City | Alexander |

Period | 9/01/22 → 12/01/22 |

### Bibliographical note

Funding Information:∗Department of Computer Science, Bar Ilan University, Ramat Gan 5290002, Israel. E-mail avi.kadria3@gmail.com. †Department of Computer Science, Bar Ilan University, Ramat Gan 5290002, Israel. E-mail liam.roditty@biu.ac.il. Supported in part by BSF grants 2016365 and 2020356. ‡Departments of Management Science and Engineering and Computer Science, Stanford University, Stanford, CA, 94305, USA. E-mail sidford@stanford.edu. Supported in part by BSF grant no. 2016365, a Microsoft Research Faculty Fellowship, NSF CAREER Award CCF-1844855, NSF Grant CCF-1955039, a PayPal research award, and a Sloan Research Fellowship §Department of Electrical Engineering and Computer Science and CSAIL, MIT, Cambridge, MA, USA. E-mail virg@mit.edu. Supported in part by NSF CAREER Award 1651838, NSF Grants CCF-1909429 and CCF-2129139, BSF grants 2016365 and 2020356, a Google Research Fellowship and a Sloan Research Fellowship. ¶Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 6997801, Israel. E-mail zwick@tau.ac.il. Supported in part by BSF grants 2016365 and 2020356.

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