发表机构
University of California, Santa Barbara; The Institute of Mathematical Sciences, HBNI; University of Bergen; New York University Shanghai（加州大学圣塔芭芭拉分校; 马哈拉施特拉邦数学科学研究所，HBNI; 卑尔根大学; 纽约大学上海校区）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对平面有向图上的有向反馈顶点集问题，提出首个单指数固定参数算法（随机化 $4.24^k$，确定性 $8.04^k$），利用欧拉型计数恒等式归约到多项式可解的有向反馈弧集。
AI 中文摘要
我们考虑平面有向图上的有向反馈顶点集问题，以解的大小 $k$ 为参数。我们给出一个具有单侧错误的随机化算法，运行时间为 $(2+\sqrt5)^k n^{O(1)}= 4.24^k n^{O(1)}$，以及一个确定性算法，运行时间为 $8.04^k n^{O(1)}$。两个算法均使用多项式空间。据我们所知，这是平面有向图上有向反馈顶点集的首个单指数固定参数算法。这与一般有向图形成对比，一般有向图的最佳已知算法运行时间为 $2^{O(k\log k)}(n+m)$，而是否存在 $2^{o(k\log k)}n^{O(1)}$ 时间的算法仍是一个重大开放问题。我们的主要工具是平面有向图的一个精确欧拉型计数恒等式。它表明每个小解必须包含大量在嵌入中入弧和出弧交替的顶点，而避免此类顶点的解可以通过归约到有向反馈弧集来计算，后者已知可通过 Lucchesi-Younger 定理在平面有向图上多项式时间求解。
英文摘要
We consider Directed Feedback Vertex Set on planar digraphs, parameterized by the solution size $k$. We give a randomized algorithm with one-sided error running in time $(2+\sqrt5)^k n^{O(1)}= 4.24^k n^{O(1)}$, and a deterministic algorithm running in time $8.04^k n^{O(1)}$. Both algorithms use polynomial space. To the best of our knowledge, these are the first single-exponential fixed-parameter algorithms for Directed Feedback Vertex Set on planar digraphs. This contrasts with general digraphs, where the best known algorithms run in time $2^{O(k\log k)}(n+m)$, and whether a $2^{o(k\log k)}n^{O(1)}$-time algorithm exists remains a major open problem. Our main tool is an exact Euler-type counting identity for plane digraphs. It shows that every small solution must carry a large share of the vertices whose in- and out-arcs alternate in the embedding, while solutions avoiding such vertices can be computed by reducing to Directed Feedback Arc Set, which is known to be solvable in polynomial time on planar digraphs via the Lucchesi-Younger theorem.
