The Feedback vertex set with the minimum size is one of Karp's 21 NP-complete problems targeted at breaking all the cycles in a graph. This problem is applicable to a broad variety of domains, including E-commerce networks, database systems, and program analysis. In reality, users are frequently most concerned with the hop-constrained cycles (i.e., cycles with a limited number of hops). For instance, in the E-commerce networks, the fraud detection team would discard cycles with a high number of hops since they are less relevant and grow exponentially in size. Thus, it is quite reasonable to investigate the feedback vertex set problem in the context of hop-constrained cycles, namely hop-constrained cycle cover problem. It is concerned with determining a set of vertices that covers all hop-constrained cycles in a given directed graph. A common method to solve this is to use a bottom-up algorithm, where it iteratively selects cover vertices into the result set. Based on this paradigm, the existing works mainly focus on the vertices orders and several heuristic strategies. In this paper, a totally opposite cover process topdown is proposed and bounds are presented on it. Surprisingly, both theoretical time complexity and practical performance are improved.

翻译：最小尺寸的反馈顶点设置为Karp 21 NP 完整的问题之一,目标是在图表中打破所有周期。 这个问题适用于广泛的领域, 包括电子商务网络、 数据库系统 以及程序分析。 事实上, 用户通常最关心的是跳动控制周期( 即跳跃次数有限的周期 ) 。 例如, 在电子商务网络中, 欺诈检测小组会丢弃周期, 跳跳次数较多, 因为跳跳次数较少, 跳跳次数成倍增长。 因此, 调查在跳动控制周期( 跳动控制周期 ) 背景下的反馈顶点设置的问题相当合理 。 这个问题适用于广泛的领域, 包括电子商务网络、 数据库系统以及程序分析 。 解决的常见方法是使用自下而上算法, 它反复地选择结果设置的顶点。 根据这种模式, 现有的工作主要侧重于悬浮命令和若干超振动战略 。 在这份论文中, 一个完全相反的顶层进程 被提出来, 一个完全相反的顶层 。