Optimal low-degree hardness of maximum independent set

We study the algorithmic task of finding a large independent set in a sparse Erdős– Rényi random graph with n vertices and average degree d . The maximum independent set is known to have size (2 \log d / d)n in the double limit n \to \infty followed by d \to \infty , but the best known polynomial-ti...

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Bibliographic Details
Published in:Mathematical statistics and learning (Online) 2022-01, Vol.4 (3), p.221-251
Main Author: Wein, Alexander S.
Format: Article
Language:English
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Summary:We study the algorithmic task of finding a large independent set in a sparse Erdős– Rényi random graph with n vertices and average degree d . The maximum independent set is known to have size (2 \log d / d)n in the double limit n \to \infty followed by d \to \infty , but the best known polynomial-time algorithms can only find an independent set of half-optimal size (\log d / d)n . We show that the class of low-degree polynomial algorithms can find independent sets of half-optimal size but no larger, improving upon a result of Gamarnik, Jagannath, and the author. This generalizes earlier work by Rahman and Virág, which proved the analogous result for the weaker class of local algorithms .
ISSN:2520-2316
2520-2324
DOI:10.4171/msl/25