PACKING AND COVERING A GIVEN DIRECTED GRAPH IN A DIRECTED GRAPH

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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 languageEnglish
Pages (from-to)43-54
Number of pages12
JournalSIAM Journal on Discrete Mathematics
Volume38
Issue number1
DOIs
StatePublished - 2024

Bibliographical note

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

Keywords

  • approximation
  • covering
  • cycle
  • packing

ASJC Scopus subject areas

  • General Mathematics

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