Optimal low-degree hardness of maximum independent set

Abstract

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.