We study reduction rules for Directed Feedback Vertex Set (DFVS) on directed graphs 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 $d$, 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. We prove that for every nowhere dense class $\mathscr{C}$ there is a function $f_\mathscr{C}(d,\epsilon)$ such that for graphs $G\in \mathscr{C}$ without induced cycles of length greater than $d$ we can compute a kernel with $f_\mathscr{C}(d,\epsilon)\cdot k^{1+\epsilon}$ vertices for any $\epsilon>0$ in time $f_\mathscr{C}(d,\epsilon)\cdot n^{O(1)}$. The most restricted classes we consider are strongly connected planar graphs without any (induced or non-induced) long cycles. We show that these classes have treewidth $O(d)$ 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 two standard rules subsumes all rules applied in the work of 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：我们研究了在无长环的有向图上定向反馈顶点集（DFVS）的约简规则。一个不含长度超过$d$的环的DFVS实例自然地对应于一个$d$-击中集实例，然而，枚举一个$n$顶点图中的所有环，然后对得到的$d$-击中集实例进行核化可能代价过高，因为仅枚举所有环就可能花费$\Omega(n^d)$的时间。我们展示了如何计算一个核，其顶点数最多为$2^dk^d$，且长度不超过$d$的诱导环数最多为$d^{3d}k^d$，其中$k$是最小定向反馈顶点集的大小。接着，我们研究了那些底层无向图具有有界扩张或是无处稠密的图类。我们证明，对于每个无处稠密类$\mathscr{C}$，存在一个函数$f_\mathscr{C}(d,\epsilon)$，使得对于不含长度大于$d$的诱导环的图$G\in \mathscr{C}$，我们可以在$f_\mathscr{C}(d,\epsilon)\cdot n^{O(1)}$的时间内，对任意$\epsilon>0$，计算出一个具有$f_\mathscr{C}(d,\epsilon)\cdot k^{1+\epsilon}$个顶点的核。我们考虑的最受限图类是不含任何（诱导或非诱导）长环的强连通平面图。我们证明这些图类的树宽为$O(d)$，因此，在无长度大于$d$的环的平面图上，DFVS可以在$2^{O(d)}\cdot n^{O(1)}$的时间内求解。最后，我们为一般DFVS提出了一种新的数据约简规则，并证明该规则与两条标准规则一起，涵盖了Bergougnoux等人的工作中为DFVS[FVS]（即由底层（无向）图的反馈顶点集数参数化的DFVS）获得多项式核所应用的所有规则。最后，我们研究了DFVS基于线性规划的近似算法。