  We study the algorithmic task of finding a large independent set in a sparse
Erd\H{o}s-R\'{e}nyi 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 Vir\'ag, which proved the analogous
result for the weaker class of local algorithms.
