Efficient improvement over Shearer in random regular graphs

Determine whether a randomized polynomial-time algorithm can find, with high probability, an independent set of size (1+ε)n log d/d in the random regular graph G(n,d), for a fixed positive ε.

Background

The paper distinguishes the algorithmic question from the existential problem: large independent sets are known to exist in the random regular graph, but known structural obstructions make efficient discovery difficult. The stated problem asks for an algorithmic improvement over Shearer’s bound by a constant factor.

References

More precisely, does there exist a randomized polynomial time algorithm that finds an independent set in the random regular graph $G(n,d)$ of size $ (1+)n(\log d)/d$, with high probability?

— Probabilistic combinatorics at exponentially small scales  (2512.15077 - Sahasrabudhe, 17 Dec 2025) in Section 3, subsection “Towards an optimal version of Shearer’s theorem”