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Low-degree threshold for independent sets in dense random graphs remains open

Determine the low-degree polynomial algorithm threshold for computing large independent sets in the Erdős–Rényi graph G(n, p) when p is a fixed constant (p = Θ(1)), by characterizing the achievable and unattainable independent-set densities for low-degree algorithms in this regime.

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Background

The authors highlight that, although Matula and Karp’s results suggest a factor-1/2 statistical–computational gap for independent sets when p = Θ(1), it is unclear how to express Karp’s algorithm within the low-degree framework.

Consequently, the precise low-degree computational threshold in the dense regime remains unresolved, motivating future work to bridge algorithmic constructions and low-degree analyses.

References

It is not clear how to describe Karp's algorithm as a low-degree polynomial and so the question of the low-degree threshold in this regime still remains open.

— The Low-Degree Hardness of Finding Large Independent Sets in Sparse Random Hypergraphs (2404.03842 - Dhawan et al., 5 Apr 2024) in Subsection ‘Concluding Remarks’