In the Directed Feedback Vertex Set (DFVS) problem, the input is a directed graph D and an integer k. The objective is to determine whether there exists a set of at most k vertices intersecting every directed cycle of D. DFVS was shown to be fixed-parameter tractable when parameterized by solution size by Chen et al. (J ACM 55(5):177–186, 2008); since then, the existence of a polynomial kernel for this problem has become one of the largest open problems in the area of parameterized algorithmics. Since this problem has remained open in spite of the best efforts of a number of prominent researchers and pioneers in the field, a natural step forward is to study the kernelization complexity of DFVS parameterized by a natural larger parameter. In this paper, we study DFVS parameterized by the feedback vertex set number of the underlying undirected graph. We provide two main contributions: a polynomial kernel for this problem on general instances, and a linear kernel for the case where the input digraph is embeddable on a surface of bounded genus.

中文翻译：

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朝向有向反馈顶点集的多项式核

在有向反馈顶点集 (DFVS) 问题中，输入是有向图 D 和整数 k。目标是确定是否存在一组至多 k 个顶点与 D 的每个有向循环相交。 当 Chen 等人通过解决方案大小进行参数化时，DFVS 被证明是固定参数易处理的。(J ACM 55(5):177–186, 2008)；从那时起，这个问题的多项式核的存在已成为参数化算法领域最大的开放问题之一。尽管该领域的许多杰出研究人员和先驱做出了最大努力，但这个问题仍然没有解决，因此自然而然的一步是研究由自然更大参数参数化的 DFVS 的核化复杂性。在本文中，我们研究了由底层无向图的反馈顶点集数参数化的 DFVS。