发表机构
State University of Campinas (Unicamp)（坎皮纳斯州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对带部分或噪声预测的最小顶点覆盖问题，提出随机多项式时间算法，实现低于2的期望近似比，匹配或接近最优渐近改进。
AI 中文摘要
我们研究了在两种非自适应离线建议模型下的最小顶点覆盖问题，其中关于一个固定最优解的预测作为输入的一部分一次性提供。这些预测形式分别由Cohen-Addad、d'Orsi、Gupta、Lee和Panigrahi（2024）以及Ghoshal、Makarychev和Makarychev（2025）独立引入。在部分预测（Partial Predictions）中，独立揭示的顶点带有正确的成员标签。对于每个足够小的固定$\varepsilon>0$，我们给出一个随机多项式时间算法，其期望近似比至多为$2-(2-o(1))\frac{\log\log(1/\varepsilon)}{\log(1/\varepsilon)}$。在噪声预测（Noisy Predictions）中，每个顶点反而接收一个相互独立的、偏差为$\varepsilon$的噪声成员标签。对于每个足够小的固定$\varepsilon>0$，我们给出一个随机多项式时间算法，其期望近似比至多为$2-(1-o(1))\frac{\log\log(1/\varepsilon)}{\log(1/\varepsilon)}$。我们的噪声预测界匹配了Aamand、Chen、Gollapudi、Silwal和Wu（2025）获得的低于2的领先渐近改进，尽管每个顶点仅使用一个噪声标签，而不是为每条关联边的每个端点使用单独的独立标签。当$\varepsilon$趋于0时，我们的两个界均成立，且严格低于2，即Khot和Regev（2008）证明的Unique Games猜想下无建议最小顶点覆盖的最优近似阈值。
英文摘要
We study the minimum Vertex Cover problem under two non-adaptive offline advice models, where predictions about a fixed optimum solution are provided once as part of the input. These forms of prediction were introduced independently by Cohen-Addad, d'Orsi, Gupta, Lee, and Panigrahi (2024) and by Ghoshal, Makarychev, and Makarychev (2025). In Partial Predictions, independently revealed vertices come with correct membership labels. For every sufficiently small fixed $\varepsilon>0$, we give a randomized polynomial-time algorithm that has expected approximation ratio at most $2-(2-o(1))\frac{\log\log(1/\varepsilon)}{\log(1/\varepsilon)}$. In Noisy Predictions, every vertex instead receives a mutually independent noisy membership label of bias $\varepsilon$. For every sufficiently small fixed $\varepsilon>0$, we give a randomized polynomial-time algorithm that has expected approximation ratio at most $2-(1-o(1))\frac{\log\log(1/\varepsilon)}{\log(1/\varepsilon)}$. Our Noisy Predictions bound matches the leading asymptotic improvement below $2$ obtained by Aamand, Chen, Gollapudi, Silwal, and Wu (2025), despite using only one noisy label per vertex rather than a separate independent label for each endpoint of each incident edge. Both our bounds hold as $\varepsilon$ goes to $0$ and are strictly below $2$, the optimal approximation threshold for minimum Vertex Cover without advice under the Unique Games Conjecture, as shown by Khot and Regev (2008).
