Improved Certificates for Independence Number in Semirandom Hypergraphs

2026-03-09

AI Summary

This work addresses the challenge of certifying tight upper bounds on the independence number of ℓ-uniform hypergraphs, which is notoriously hard to approximate in the worst case. By designing robust Sum-of-Squares (SoS) certificates within a semi-random model, the authors leverage degree-2ℓ SoS relaxations, pseudo-expectation analysis, tensor methods, and a novel concentration inequality for random chaotic matrices to eliminate the logarithmic factors present in prior results. This yields the first tight, efficiently certifiable upper bound of \(O(\sqrt{n}/p^{1/\ell})\), which nearly matches the conjectured computational threshold in terms of both \(n\) and \(p\). The approach applies to hypergraphs of odd uniformity and successfully recovers planted \(r\)-colorable sub-hypergraphs even under strong adversarial perturbations.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Security and Privacy: Applications of cryptography

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 $n^{1-\epsilon}$ 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 $O(\sqrt{n}.\mathrm{polylog}(n)/p^{1/(\ell/2)})$. 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 $O(\sqrt{n}/p^{1/\ell})$, 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.

Problem

• independence number
• semirandom hypergraphs
• certificates
• uniform hypergraphs
• computational thresholds

Innovation

• Sum-of-Squares certificates
• semirandom hypergraphs
• independence number
• tensor analysis
• matrix concentration inequalities