半随机超图中独立数的改进证书
Improved Certificates for Independence Number in Semirandom Hypergraphs
AI总结：
针对半随机超图独立数上界认证问题，通过度-2ℓ SoS松弛和随机混沌矩阵范数分析，消除了n的对数因子，几乎达到最优阈值O(√n/p^{1/ℓ})，并构造了低度不可区分的安静种植分布。
摘要：
我们研究了在半随机模型中高效认证ℓ-均匀超图独立数上界的问题。这是一个众所周知的难题，在最坏情况下，高效算法无法在n^{1-ε}因子内逼近独立数。一个经典的归约到图的情况给出了弱的O(√(n/p))界，而谱证书[GKM22]达到了O(√n·polylog(n)/p^{2/ℓ})。在这项工作中，我们证明了更紧的界，消除了n中的对数因子，并几乎达到了最优阈值O(√n/p^{1/ℓ})。我们还展示了匹配的低次多项式下界。我们的证书是通过证明到算法的范式设计的，使用了度-2ℓ平方和(SoS)松弛。技术上具有挑战性的情况是奇元超图，其中我们采用基于张量的分析，将问题简化为随机混沌矩阵的算子范数界。先前的界[AMP21,RT23]具有对数依赖性，我们使用最近的矩阵集中不等式[BBvH23,BLNvH25]将其去除；我们相信这可能在其它超图问题中有用。由于我们在SoS框架中部署证书，这些界对于单调对手仍然成立。此外，我们构造了一个“安静”的种植分布，支持大小为k=o(√n/p^{1/ℓ})的独立集，该分布与随机超图在低次意义下不可区分。在此之前，即使在图的情况下，稀疏区域中构造安静种植分布的问题也是开放的[JPR+22,Pot22]。这与恢复种植独立集形成对比，后者的阈值为k ≳ √n/p^{1/(2(ℓ-1))}（与并发工作[FS26]中的下界匹配）。作为应用，我们的证书与r-染色系统的SoS松弛结合，可以在[LPR25]的强对手下恢复种植的r-可染色子超图。
英文摘要：
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].
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