Improved Certificates for Independence Number in Semirandom Hypergraphs
This work advances the theoretical understanding of certification in hypergraph problems, offering nearly optimal bounds and resolving open questions about quiet planted distributions for sparse hypergraphs.
The authors provide improved certificates for upper bounds on the independence number of ℓ-uniform hypergraphs in semirandom models, eliminating logarithmic factors and nearly attaining the optimal threshold of O(√n / p^{1/ℓ}). They also construct a quiet planted distribution that is low-degree indistinguishable from random hypergraphs for independent sets of size o(√n / p^{1/ℓ}).
We study the problem of efficiently certifying upper bounds on independence number of $\ell$-uniform hypergraphs in semirandom models. This is a notoriously hard problem, with efficient algorithms failing to approximate the independence number within an $n^{1-ε}$ factor in worst-case. A folklore reduction to graph case yields a weak $O(\sqrt{n/p})$ bound, and spectral certificates[GKM22] achieve $O(\sqrt{n}.polylog(n)/p^{2/\ell})$. In this work, we prove sharper bounds that eliminate logarithmic factors in $n$ and nearly attain the optimal threshold of $O(\sqrt{n}/p^{1/\ell})$. We also show matching low-degree polynomial lower bounds. Our certificates are designed using the proofs-to-algorithms paradigm via degree-$2\ell$ Sum-of-Squares(SoS) relaxation. The technically challenging case is odd-arity hypergraphs, where we employ a tensor-based analysis reducing the problem to bounding operator norm of random chaos matrices. Previous bounds[AMP21,RT23] have a logarithmic dependence, which we remove using recent matrix concentration inequalities[BBvH23,BLNvH25]; we believe this maybe useful in other hypergraph problems. Since we deploy our certificates in SoS framework, the bounds continue to hold for monotone adversaries. Additionally, we construct a 'quiet' planted distribution supported on independent sets of size $k=o(\sqrt{n}/p^{1/\ell})$ that is low-degree indistinguishable from random hypergraphs. Prior to this work, the problem of constructing a quiet planted distribution in sparse regimes was open even for graphs[JPR+22,Pot22]. This is in contrast to recovering a planted independent set, where the threshold is $k \gtrsim \sqrt{n}/p^{1/2(\ell-1)}$ (matching lower bounds in a concurrent work[FS26]). As application, our certificates combine with an SoS relaxation of an r-coloring system to recover a planted r-colorable subhypergraph under strong adversaries of [LPR25].