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

Prove that for every epsilon > 0 and integer r ≥ 2, no polynomial-time algorithm can, with high probability, find a balanced independent set in the random balanced r-uniform r-partite hypergraph H(r, n, d/n^{r−1}) whose density is at least (1+epsilon)·((1/(r−1)·log d)/d)^{1/(r−1)} for sufficiently large d and n.

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Background

The authors extend their low-degree threshold results to balanced independent sets in random r-uniform r-partite hypergraphs and show a matching barrier for low-degree algorithms at the same density scale as in ordinary hypergraphs. They conjecture that this barrier is fundamental for all polynomial-time algorithms.

This conjecture parallels the r = 2 bipartite graph case recently studied, positing a universal r{-1/(r−1)} multiplicative gap between the statistical optimum and computationally achievable densities for balanced independent sets.

References

We also conjecture that this statistical-computational gap persists for polynomial-time algorithms (a version of this conjecture for $r = 2$ appeared in ).

For any $ > 0$ and integer $r\geq 2$, there are $d,n \in N$ sufficiently large such that there is no polynomial-time algorithm that finds a balanced independent set in $\mathcal{H}\left(r, n, d/n{r-1}\right)$ 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., 5 Apr 2024) in Conjecture 2 (label ‘conjecture: balanced hypergraph SCG’), Introduction