  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.
