## Abstract

For every fixed k ≥ 4, it is proved that if an n-vertex directed graph has at most t pairwise arc-disjoint directed k-cycles, then there exists a set of at most ^{2}_{3} kt + o(n^{2}) arcs that meets all directed k-cycles and that the set of k-cycles admits a fractional cover of value at most ^{2}_{3} kt. It is also proved that the ratio ^{2}_{3} k cannot be improved to a constant smaller than ^{k}_{2}. For k = 5 the constant 2k/3 is improved to 25/8 and for k = 3 it was recently shown by Cooper et al. [European J. Combin., 101 (2022), 103462] that the constant can be taken to be 9/5. The result implies a deterministic polynomial time _{3}^{2} k-approximation algorithm for the directed k-cycle cover problem, improving upon a previous (k-1)-approximation algorithm of Kortsarz, Langberg, and Nutov, [SIAM J. Discrete Math., 24 (2010), pp. 255-269]. More generally, for every directed graph H we introduce a graph parameter f(H) for which it is proved that if an n-vertex directed graph has at most t pairwise arc-disjoint H-copies, then there exists a set of at most f(H)t + o(n^{2}) arcs that meets all H-copies and that the set of H-copies admits a fractional cover of value at most f(H)t. It is shown that for almost all H it holds that f(H) ≈ |E(H)|/2 and that for every k-vertex tournament H it holds that f(H) ≤ ⌊k^{2}/4⌋.

Original language | English |
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Pages (from-to) | 43-54 |

Number of pages | 12 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 38 |

Issue number | 1 |

DOIs | |

State | Published - 2024 |

### Bibliographical note

Publisher Copyright:© 2024 Society for Industrial and Applied Mathematics.

## Keywords

- approximation
- covering
- cycle
- packing

## ASJC Scopus subject areas

- General Mathematics