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 → ∞ followed by d → ∞ , 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 .