Vertex Cover Problem with Advice
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).