Improved Certificates for Independence Number in Semirandom Hypergraphs

Abstract: We study the problem of efficiently certifying upper bounds on the independence number of $\ell$-uniform hypergraphs. This is a notoriously hard problem, with efficient algorithms failing to approximate the independence number within $n<sup>{1-ε}$ factor in the worst case [Has99, Zuc07]. We study the problem in random and semirandom hypergraphs. There is a folklore reduction to the graph case, achieving a certifiable bound of $O(\sqrt{n/p})$. More recently, the work [GKM22] improved this by constructing spectral certificates that yield a bound of $O(\sqrt{n}.\mathrm{polylog}(n)/p<sup>{1/(\ell/2)})$. We make two key improvements: firstly, we prove sharper bounds that get rid of pesky logarithmic factors in $n$, and nearly attain the conjectured optimal (in both $n$ and $p$) computational threshold of $O(\sqrt{n}/p<sup>{1/\ell})$, and secondly, we design robust Sum-of-Squares (SoS) certificates, proving our bounds in the more challenging semirandom hypergraph model. Our analysis employs the proofs-to-algorithms paradigm [BS16, FKP19] in showing an upper bound for pseudo-expectation of degree-$2\ell$ SoS relaxation of the natural polynomial system for maximum independent set. The challenging case is odd-arity hypergraphs, where we employ a tensor-based analysis that reduces the problem to proving bounds on a natural class of random chaos matrices associated with $\ell$-uniform hypergraphs. Previous bounds [AMP21, RT23] have a logarithmic dependence, which we remove by leveraging recent progress on matrix concentration inequalities [BBvH23, BLNvH25]; we believe these may be useful in other hypergraph problems. As an application, we show our improved certificates can be combined with an SoS relaxation of a natural $r$-coloring polynomial system to recover an arbitrary planted $r$-colorable subhypergraph in a semirandom model along the lines of [LPR25], which allows for strong adversaries.