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Sparse random graph independent set intractability conjecture

Prove that for Erdős–Rényi graphs G(n, d/n) with sufficiently large d, no polynomial-time algorithm can, with high probability, find an independent set whose density is at least (1+epsilon)·(log d/d) for any fixed epsilon > 0, establishing a factor-1/2 statistical–computational gap.

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Background

In the graph case (r = 2), the maximum independent set in the sparse regime has density approximately 2·(log d)/d, but it is widely conjectured that polynomial-time algorithms cannot reach densities exceeding (log d)/d up to a (1+epsilon) factor, implying a factor-1/2 gap.

This conjecture motivates the hypergraph generalizations studied in the paper and provides a baseline for the r-uniform results.

References

It is conjectured that no polynomial time algorithm can find an independent set of density $(1+)\log d/d$ with high probability for any $\varepsilon>0$, i.e., there is a statistical-computational gap of a multiplicative factor of $1/2$.

— The Low-Degree Hardness of Finding Large Independent Sets in Sparse Random Hypergraphs (2404.03842 - Dhawan et al., 5 Apr 2024) in Introduction (Independent Sets in Graphs and Hypergraphs)