We study the algorithmic tractability of finding large independent sets in dense random hypergraphs. In the sparse regime, much of the natural algorithms can be formulated within either the local or the low-degree polynomial (LDP) framework, and a rich literature has subsequently identified nearly sharp algorithmic thresholds within these classes by exploiting their stability. In the dense setting, however, the algorithmic paradigms are fundamentally different: they are online and thus need not be stable. Perhaps more crucially, even for the classical Erdős-Rényi random graph $G(n,p)$, LDPs are conjectured to fail in the 'easy' regime accessible to online algorithms, thereby challenging their viability for dense models. Our focus is on two models: (i) finding large independent sets in dense $r$-uniform Erdős-Rényi hypergraphs, and (ii) the more challenging problem of finding large $γ$-balanced independent sets in dense $r$-uniform $r$-partite hypergraphs, where the $i$-th coordinate of $γ\in\mathbb{Q}^r$ specifies the proportion of vertices from $V_i$ in the independent set. For both models, we pinpoint the size of the largest independent set and design online algorithms that achieve a multiplicative approximation factor of $r^{1/(r-1)}$ in the uniform and $(\max_i γ_i)^{-1/(r-1)}$ in the $r$-partite model. Furthermore, we establish matching algorithmic lower bounds, showing that these computational gaps are sharp: no online algorithms can breach these gaps.

翻译：我们研究了稠密随机超图中寻找大独立集的算法可解性。在稀疏情形下，许多自然算法可纳入局部或低度多项式（LDP）框架，已有丰富文献通过利用其稳定性在这些框架内确定了接近锐利的算法阈值。然而在稠密设定下，算法范式根本不同：算法是在线的，因而不必具有稳定性。或许更关键的是，即使对于经典埃尔迪什-雷尼随机图$G(n,p)$，LDP也被推测会在在线算法可解的"简单"区域失效，从而对其在稠密模型中的可行性提出挑战。我们聚焦两类模型：(i) 在稠密$r$一致埃尔迪什-雷尼超图中寻找大独立集，以及(ii) 更具挑战性的问题——在稠密$r$一致$r$部分超图中寻找大$\gamma$平衡独立集，其中$\gamma\in\mathbb{Q}^r$的第$i$个坐标指定独立集中来自$V_i$的顶点比例。对于两类模型，我们精确确定了最大独立集的大小，并设计了在线算法，在均匀模型中达到$r^{1/(r-1)}$的乘法近似因子，在$r$部分模型中达到$(\max_i \gamma_i)^{-1/(r-1)}$的近似因子。此外，我们建立了匹配的算法下界，证明这些计算间隙是锐利的：没有在线算法能突破这些间隙。