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The Feedback vertex set with the minimum size is one of Karp's 21 NP-completeproblems targeted at breaking all the cycles in a graph. This problem isapplicable to a broad variety of domains, including E-commerce networks,database systems, and program analysis. In reality, users are frequently mostconcerned with the hop-constrained cycles (i.e., cycles with a limited numberof hops). For instance, in the E-commerce networks, the fraud detection teamwould discard cycles with a high number of hops since they are less relevantand grow exponentially in size. Thus, it is quite reasonable to investigate thefeedback vertex set problem in the context of hop-constrained cycles, namelyhop-constrained cycle cover problem. It is concerned with determining a set ofvertices that covers all hop-constrained cycles in a given directed graph. Acommon method to solve this is to use a bottom-up algorithm, where ititeratively selects cover vertices into the result set. Based on this paradigm,the existing works mainly focus on the vertices orders and several heuristicstrategies. In this paper, a totally opposite cover process topdown is proposedand bounds are presented on it. Surprisingly, both theoretical time complexityand practical performance are improved.

TDB: Breaking All Hop-Constrained Cycles in Billion-Scale Directed Graphs