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.

References

In light of this result and the aforementioned statistical-computational gap conjecture of independent sets in random graphs, we make the following analogous conjecture for hypergraphs.

For any fixed $ > 0$ and integer $r\geq 2$, there are $d,n \in N$ sufficiently large such that there is no polynomial-time algorithm that finds an independent set in $\mathcal{H}_r\left(n, d/\binom{n-1}{r-1}\right)$ of density at least $(1+)\left(\frac{1}{r-1}\cdot\frac{\log d}{d}\right){1/(r-1)}$ with high probability.

— The Low-Degree Hardness of Finding Large Independent Sets in Sparse Random Hypergraphs  (2404.03842 - Dhawan et al., 2024) in Conjecture 1 (label ‘conjecture: hypergraph SCG’), Introduction

It is natural to ask whether the local statement remains true under the weaker assumption of linearity.

\begin{problem}\label{problem: local linear demands} Let $k \ge 3$ be an integer, and let $\eps > 0$ be arbitrary. Does there exist $\delta_0 \in \N$ such that the following holds? If $\cH$ is a $k$-uniform linear hypergraph of minimum degree at least $\delta_0$, and $\phi$ is a demand function for $\cH$ satisfying, for every $v \in V(\cH)$,

\phi(v) \le (1-\eps)\left(\frac{1}{k-1} \, \frac{\log \deg(v)}{\deg(v)}\right){\frac{1}{k-1},

must $\cH$ admit a $\phi$-coloring? \end{problem}

— Random independent sets in uncrowded hypergraphs  (2610.08707 - Dhawan et al., 6 Oct 2026) in Problem 2, Section 5 (Concluding remarks)