We study reduction rules for Directed Feedback Vertex Set (DFVS) on instances without long cycles. A DFVS instance without cycles longer than $d$ naturally corresponds to an instance of $d$-Hitting Set, however, enumerating all cycles in an $n$-vertex graph and then kernelizing the resulting $d$-Hitting Set instance can be too costly, as already enumerating all cycles can take time $\Omega(n^d)$. We show how to compute a kernel with at most $2^dk^d$ vertices and at most $d^{3d}k^d$ induced cycles of length at most $d$ (which however, cannot be enumerated efficiently), where $k$ is the size of a minimum directed feedback vertex set. We then study classes of graphs whose underlying undirected graphs have bounded expansion or are nowhere dense; these are very general classes of sparse graphs, containing e.g. classes excluding a minor or a topological minor. We prove that for such classes without induced cycles of length greater than $d$ we can compute a kernel with $O_d(k)$ and $O_{d,\epsilon}(k^{1+\epsilon})$ vertices for any $\epsilon>0$, respectively, in time $O_d(n^{O(1)})$ and $O_{d,\epsilon}(n^{O(1)})$, respectively. The most restricted classes we consider are strongly connected planar graphs without any (induced or non-induced) long cycles. We show that these have bounded treewidth and hence DFVS on planar graphs without cycles of length greater than $d$ can be solved in time $2^{O(d)}\cdot n^{O(1)}$. We finally present a new data reduction rule for general DFVS and prove that the rule together with a few standard rules subsumes all the rules applied by Bergougnoux et al. to obtain a polynomial kernel for DFVS[FVS], i.e., DFVS parameterized by the feedback vertex set number of the underlying (undirected) graph. We conclude by studying the LP-based approximation of DFVS.

翻译：我们研究针对不含长圈的有向反馈顶点集（DFVS）实例的约简规则。不含长度超过$d$的圈的有向反馈顶点集实例自然对应一个$d$-击中集实例，然而，枚举$n$顶点图中的所有圈并对所得$d$-击中集实例进行核化可能代价过高，因为仅枚举所有圈就需要时间$\Omega(n^d)$。我们展示了如何计算一个规模不超过$2^d k^d$个顶点且包含至多$d^{3d}k^d$个长度不超过$d$的诱导环（但这些环无法高效枚举）的核，其中$k$是最小有向反馈顶点集的大小。随后我们研究底图无向图具有有界扩展性或无处稠密的图类——这些是非常一般的稀疏图类，包含例如排除某个子式或拓扑子式的图类。我们证明，对于此类不含长度超过$d$的诱导环的图类，我们能在时间$O_d(n^{O(1)})$和$O_{d,\epsilon}(n^{O(1)})$内分别计算规模为$O_d(k)$和$O_{d,\epsilon}(k^{1+\epsilon})$（对任意$\epsilon>0$）的核。我们考虑的最受限制图类是强连通平面图，且不含任何（诱导或非诱导）长环。我们证明这些图具有有界树宽，因此对于不含长度超过$d$的环的平面图，可在时间$2^{O(d)}\cdot n^{O(1)}$内求解有向反馈顶点集问题。最后，我们提出一个适用于一般有向反馈顶点集的新数据约简规则，并证明该规则结合若干标准规则可涵盖Bergougnoux等人为获得有向反馈顶点集问题[基于反馈顶点集参数化，即参数化底层（无向）图的反馈顶点集数]的多项式核所使用的所有约简规则。我们通过研究基于线性规划的有向反馈顶点集近似算法来结束本文。