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Packing and Covering a Given Directed Graph in a Directed Graph
SIAM Journal on Discrete Mathematics ( IF 0.8 ) Pub Date : 2024-01-04 , DOI: 10.1137/22m1534134 Raphael Yuster 1
SIAM Journal on Discrete Mathematics ( IF 0.8 ) Pub Date : 2024-01-04 , DOI: 10.1137/22m1534134 Raphael Yuster 1
Affiliation
SIAM Journal on Discrete Mathematics, Volume 38, Issue 1, Page 43-54, March 2024.
Abstract. For every fixed [math], it is proved that if an [math]-vertex directed graph has at most [math] pairwise arc-disjoint directed [math]-cycles, then there exists a set of at most [math] arcs that meets all directed [math]-cycles and that the set of [math]-cycles admits a fractional cover of value at most [math]. It is also proved that the ratio [math] cannot be improved to a constant smaller than [math]. For [math] the constant [math] is improved to [math] and for [math] it was recently shown by Cooper et al. [European J. Combin., 101 (2022), 103462] that the constant can be taken to be [math]. The result implies a deterministic polynomial time [math]-approximation algorithm for the directed [math]-cycle cover problem, improving upon a previous [math]-approximation algorithm of Kortsarz, Langberg, and Nutov, [SIAM J. Discrete Math., 24 (2010), pp. 255–269]. More generally, for every directed graph [math] we introduce a graph parameter [math] for which it is proved that if an [math]-vertex directed graph has at most [math] pairwise arc-disjoint [math]-copies, then there exists a set of at most [math] arcs that meets all [math]-copies and that the set of [math]-copies admits a fractional cover of value at most [math]. It is shown that for almost all [math] it holds that [math] and that for every [math]-vertex tournament [math] it holds that [math].
中文翻译：
在有向图中包装和覆盖给定的有向图
SIAM 离散数学杂志，第 38 卷，第 1 期，第 43-54 页，2024 年 3 月。
摘要。对于每个固定的[math]，证明如果[math]-顶点有向图至多具有[math]个成对弧不相交的有向[math]-圈，则存在一组至多[math]个弧满足所有有向[数学]循环，并且[数学]循环集合最多允许[数学]值的分数覆盖。还证明了比率[math]不能改进为小于[math]的常数。对于 [math]，常数 [math] 被改进为 [math]，而对于 [math]，Cooper 等人最近表明了这一点。[European J. Combin., 101 (2022), 103462] 该常数可以视为[数学]。结果意味着有向循环覆盖问题的确定性多项式时间近似算法，改进了 Kortsarz、Langberg 和 Nutov 之前的数学近似算法，[SIAM J. Discrete Math., 24 (2010)，第 255–269 页]。更一般地，对于每个有向图 [math]，我们引入一个图参数 [math]，证明如果 [math]-顶点有向图最多具有 [math] 成对弧不相交 [math]-副本，则存在一组最多[数学]弧满足所有[数学]副本，并且[数学]副本集最多允许[数学]值的分数覆盖。结果表明，对于几乎所有的[数学]，它都保持[数学]，并且对于每个[数学]-顶点锦标赛[数学]，它都保持[数学]。
更新日期：2024-01-05
Abstract. For every fixed [math], it is proved that if an [math]-vertex directed graph has at most [math] pairwise arc-disjoint directed [math]-cycles, then there exists a set of at most [math] arcs that meets all directed [math]-cycles and that the set of [math]-cycles admits a fractional cover of value at most [math]. It is also proved that the ratio [math] cannot be improved to a constant smaller than [math]. For [math] the constant [math] is improved to [math] and for [math] it was recently shown by Cooper et al. [European J. Combin., 101 (2022), 103462] that the constant can be taken to be [math]. The result implies a deterministic polynomial time [math]-approximation algorithm for the directed [math]-cycle cover problem, improving upon a previous [math]-approximation algorithm of Kortsarz, Langberg, and Nutov, [SIAM J. Discrete Math., 24 (2010), pp. 255–269]. More generally, for every directed graph [math] we introduce a graph parameter [math] for which it is proved that if an [math]-vertex directed graph has at most [math] pairwise arc-disjoint [math]-copies, then there exists a set of at most [math] arcs that meets all [math]-copies and that the set of [math]-copies admits a fractional cover of value at most [math]. It is shown that for almost all [math] it holds that [math] and that for every [math]-vertex tournament [math] it holds that [math].
中文翻译：
在有向图中包装和覆盖给定的有向图
SIAM 离散数学杂志，第 38 卷，第 1 期，第 43-54 页，2024 年 3 月。
摘要。对于每个固定的[math]，证明如果[math]-顶点有向图至多具有[math]个成对弧不相交的有向[math]-圈，则存在一组至多[math]个弧满足所有有向[数学]循环，并且[数学]循环集合最多允许[数学]值的分数覆盖。还证明了比率[math]不能改进为小于[math]的常数。对于 [math]，常数 [math] 被改进为 [math]，而对于 [math]，Cooper 等人最近表明了这一点。[European J. Combin., 101 (2022), 103462] 该常数可以视为[数学]。结果意味着有向循环覆盖问题的确定性多项式时间近似算法，改进了 Kortsarz、Langberg 和 Nutov 之前的数学近似算法，[SIAM J. Discrete Math., 24 (2010)，第 255–269 页]。更一般地，对于每个有向图 [math]，我们引入一个图参数 [math]，证明如果 [math]-顶点有向图最多具有 [math] 成对弧不相交 [math]-副本，则存在一组最多[数学]弧满足所有[数学]副本，并且[数学]副本集最多允许[数学]值的分数覆盖。结果表明，对于几乎所有的[数学]，它都保持[数学]，并且对于每个[数学]-顶点锦标赛[数学]，它都保持[数学]。
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