发表机构
Grainger College of Engineering, University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校格里尔工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究通过AI辅助证明了强密度多面体的极点性质，由此得到反馈顶点集的多项式时间迭代取整2-近似算法，填补了该问题2-近似算法的方法空白。
AI 中文摘要
我们研究反馈顶点集问题（FVS）：输入为无向图G=(V,E)，目标是找到最小基数（加权情况下为最小代价）的顶点子集S⊆V，使得G-S无环。20世纪90年代中期，Bafna、Berman和Fujito（1995）以及Becker和Geiger（1996）通过局部比率法提出了2-近似算法，该近似比在独特游戏猜想（UGC）下是紧的。Chudak、Goemans、Hochbaum和Williamson（1998）后来将局部比率算法通过线性规划（LP）松弛解释为原始-对偶算法。所有已知的FVS的2-近似算法均基于局部比率和原始-对偶方法，为获得新的LP取整算法，Fiorini（2021）提出猜想：Chudak等人开发的强密度多面体具有极点性质，即该LP的每个基本可行解都存在一个变量取值至少为1/2。我们证明了该猜想，还研究了相关的强边密度多面体，并证明其具有相同的极点性质。该多面体的优势在于它允许多项式时间的分离神谕和紧凑的扩展公式。这些结果催生了多项式时间的迭代取整2-近似算法。该极点性质的证明具有独立的技术价值，证明中的关键思想由AI工具提供。
英文摘要
We consider the Feedback Vertex Set problem (FVS): the input is an undirected graph $G=(V,E)$ and the goal is to find a minimum-cardinality (or a min-cost in the weighted case) subset $S \subseteq V$ of vertices such that $G-S$ has no cycles. A $2$-approximation via the local-ratio method was developed in the mid 90's by Bafna, Berman and Fujito (1995) and by Becker and Geiger (1996), and this approximation ratio is tight under UGC. The local-ratio algorithms were later interpreted as primal-dual algorithms via an LP relaxation by Chudak, Goemans, Hochbaum, and Williamson (1998). All known $2$-approximation algorithms for FVS have been via local-ratio and primal-dual methods, and in a quest to obtain a new LP rounding algorithm, it was conjectured (Fiorini 2021) that the Strong-Density polyhedron developed by Chudak, Goemans, Hochbaum, and Williamson has an extreme point property: every basic feasible solution to the LP has a variable with value at least $1/2$. We prove this conjecture. We also consider a related Strong-Edge-Density polyhedron and show the same extreme point property. The advantage of this polyhedron is that it admits a polynomial-time separation oracle and also a compact extended formulation. These results lead to polynomial-time iterative rounding $2$-approximation algorithms. The proof of the extreme point property is of independent technical interest and key ideas in the proof were suggested by AI tools.
