Problem
Research questions and friction points this paper is trying to address.
Feedback Vertex Set
approximation algorithm
LP relaxation
An iterative rounding $2$-approximation for Feedback Vertex Set via AI-assisted proof of an extreme point property
🤖 AI Summary
该研究解决了反馈顶点集问题，通过AI辅助证明了一个极点性质，并提出了一个基于迭代舍入的2-近似算法。
📝 Abstract
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.
Innovation
Methods, ideas, or system contributions that make the work stand out.
AI-assisted proof
extreme point property
iterative rounding
polynomial-time algorithm
feedback vertex set
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