Polynomial-time independent sets in random graphs

Determine whether a polynomial-time algorithm can find, with high probability, an independent set of size at least (1+ε)(n log d)/d in the sparse random graph G_{n,p} regime described in the paper, for every fixed ε>0.

Background

The paper discusses the algorithmic barrier for finding large independent sets in the binomial random graph G_{n,p}, where d=np satisfies d=ω(log n) and d=o(n{1/3}). It states that a simple polynomial-time algorithm finds an independent set of size approximately (1−ε)(n log d)/d with high probability, whereas nonconstructive methods can achieve nearly twice this size. The unresolved issue is whether the factor can be improved from 1−ε to 1+ε by a polynomial-time algorithm.

References

But in 1976, Karp essentially asked what happens if we replace the factor $1-$ by $1+$. (In fact, Karp asked this in the dense case $p=\frac12$, but the problem is well founded and is just as difficult in sparser regimes for $p=p(n)$.)

— The hard-core model in graph theory  (2501.03379 - Davies et al., 6 Jan 2025) in Section “Barriers”