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 23 kt + o(n2) arcs that meets all directed k-cycles and that the set of k-cycles admits a fractional cover of value at most 23 kt. It is also proved that the ratio 23 k cannot be improved to a constant smaller than k2. 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 32 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(n2) 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) ≤ ⌊k2/4⌋.
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