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Vertex Cover Problem with Advice
Data Structures and Algorithms · Submitted 2 Oct 2026
abstract compiled · 2026-10-06 · no language-model review
The authors' main claim 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).
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The abstract, sentence by sentence
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).
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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. | contribution 0.71 | 0.66 | 0.05 | asserted 0.93 |
| s1 | These forms of prediction were introduced independently by Cohen-Addad, d'Orsi, Gupta, Lee, and Panigrahi (2024) and by Ghoshal, Makarychev, and Makarychev (2025). | background 1.00 | 0.03 | 0.02 | asserted 1.00 |
| s2 | In Partial Predictions, independently revealed vertices come with correct membership labels. | background 0.55 | 0.10 | 0.03 | asserted 1.00 |
| s3 | For every sufficiently small fixed $\varepsilon>0$, we give a randomized polynomial-time algorithm that has expected approximation ratio at most $2-(2-o(1))\frac{\log\log(1/\varepsilon)}{\log(1/\varepsilon)}$. | result 0.96 | 0.96 | 0.30 | asserted 1.00 |
| s4 | In Noisy Predictions, every vertex instead receives a mutually independent noisy membership label of bias $\varepsilon$. | method 0.68 | 0.37 | 0.04 | asserted 0.99 |
| s5 | For every sufficiently small fixed $\varepsilon>0$, we give a randomized polynomial-time algorithm that has expected approximation ratio at most $2-(1-o(1))\frac{\log\log(1/\varepsilon)}{\log(1/\varepsilon)}$. | result 0.98 | 0.96 | 0.25 | asserted 1.00 |
| s6 | 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. | interpretation 0.63 | 0.77 | 0.16 | asserted 1.00 |
| s7 | 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). | interpretation 0.83 | 0.57 | 0.33 | asserted 1.00 |
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Original abstract
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).
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