Optimal Low-Degree Hardness of Maximum Independent Set

TL;DR

This paper demonstrates that low-degree polynomial algorithms can find large independent sets in sparse random graphs up to the known optimal size, but no larger, highlighting their limitations.

Contribution

It establishes the optimality of low-degree polynomial algorithms for finding maximum independent sets in sparse Erdős-Rényi graphs, extending previous results to this class.

Findings

Low-degree polynomial algorithms achieve half-optimal independent set size.

No low-degree polynomial algorithm can surpass the half-optimal size.

Results generalize prior work on local algorithms.

Abstract

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.