Improved Certificates for Independence Number in Semirandom Hypergraphs

Abstract

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].