Statistical–computational gap conjecture for independent sets in sparse r-uniform hypergraphs

Prove that for every fixed epsilon > 0 and integer r ≥ 2, no polynomial-time algorithm can, with high probability, find an independent set in the sparse Erdős–Rényi r-uniform hypergraph H_r(n, d/\binom{n-1}{r-1}) whose density is at least (1+epsilon)·((1/(r−1)·log d)/d)^{1/(r−1)} for sufficiently large d and n.

Background

The paper proves sharp achievability and impossibility thresholds for low-degree polynomial algorithms on sparse Erdős–Rényi r-uniform hypergraphs, establishing a computational barrier at density ((1/(r−1)·log d)/d)^{1/(r−1)}. The authors conjecture that this barrier persists for all polynomial-time algorithms, mirroring the well-studied graph case (r = 2) where a factor-1/2 statistical–computational gap is conjectured.

This conjecture asserts a universal hardness phenomenon: even though the maximum independent set has density approximately ((r/(r−1))·(log d)/d)^{1/(r−1)}, polynomial-time algorithms cannot reach densities exceeding ((1/(r−1))·(log d)/d)^{1/(r−1)} up to a (1+epsilon) factor, for sufficiently large n and d.