Optimal Low-Degree Hardness of Maximum Independent Set
This provides theoretical limits for efficient algorithms in combinatorial optimization, but it is incremental as it generalizes earlier work to a broader class of algorithms.
The paper tackles the problem of finding large independent sets in sparse random graphs, showing that low-degree polynomial algorithms can achieve at most half-optimal size, improving upon prior results for local algorithms.
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.