AI 中文总结
本文解决了平面有向反馈顶点集问题的多项式核开放问题，提出确定性核算法，通过结构归约、平面偶图压缩和有向割论证，得到$O(k^{66}\log^2 k)$大小的核。
AI 中文摘要
有向反馈顶点集问题（DFVS）询问是否可以通过删除至多$k$个顶点使一个有向图成为无环图。DFVS是否 admits 以$k$为参数的多项式核是核化领域的一个重大开放问题，即使对于平面有向图也是如此。我们通过给出一个具有$O(k^{66}\log^2 k)$个顶点和弧的确定性核来解决平面情形。我们的算法分三个阶段进行。首先，我们对输入有向图应用结构归约规则，限制有向面的数量和某些特殊顶点。其次，我们转向平面偶图，其中顶点删除对应于添加反向弧组以使每个弱连通分量成为强连通。第一阶段的结构界在偶图中产生一个小的保留顶点集。然后，我们通过识别具有相同距离记录的顶点来压缩偶图实例，这些距离记录来自该保留顶点集。主要的技术贡献是一个有向割论证，表明这种识别保持了可行性。最后，我们通过$3$-CNF编码和平面图构造将多项式大小的偶图实例转换回平面有向反馈顶点集的一个实例。
英文摘要
The Directed Feedback Vertex Set problem (DFVS) asks whether a digraph can be made acyclic by deleting at most $k$ vertices. Whether DFVS admits a polynomial kernel parameterized by $k$ is a major open problem in kernelization, even for planar digraphs. We resolve the planar case by giving a deterministic kernel with $O(k^{66}\log^2 k)$ vertices and arcs. Our algorithm proceeds in three stages. First, we apply structural reduction rules to the input digraph, bounding the number of directed faces and some special vertices. Second, we pass to the planar dual, where vertex deletion corresponds to adding groups of reverse arcs to make each weakly connected component strongly connected. The structural bounds in the first stage yield a small retained vertex set in the dual. We then compress the dual instance by identifying vertices with the same distance records from this retained vertex set. The main technical contribution is a directed-cut argument showing that this identification preserves feasibility. Finally, we transform the polynomial-size dual instance back into an instance of Planar Directed Feedback Vertex Set via a $3$-CNF encoding and a planar graph construction.
