We study the algorithmic task of finding large independent sets in Erdos-Renyi $r$-uniform hypergraphs on $n$ vertices having average degree $d$. Krivelevich and Sudakov showed that the maximum independent set has density $\left(\frac{r\log d}{(r-1)d}\right)^{1/(r-1)}$. We show that the class of low-degree polynomial algorithms can find independent sets of density $\left(\frac{\log d}{(r-1)d}\right)^{1/(r-1)}$ but no larger. This extends and generalizes earlier results of Gamarnik and Sudan, Rahman and Virag, and Wein on graphs, and answers a question of Bal and Bennett. We conjecture that this statistical-computational gap holds for this problem. Additionally, we explore the universality of this gap by examining $r$-partite hypergraphs. A hypergraph $H=(V,E)$ is $r$-partite if there is a partition $V=V_1\cup\cdots\cup V_r$ such that each edge contains exactly one vertex from each set $V_i$. We consider the problem of finding large balanced independent sets (independent sets containing the same number of vertices in each partition) in random $r$-partite hypergraphs with $n$ vertices in each partition and average degree $d$. We prove that the maximum balanced independent set has density $\left(\frac{r\log d}{(r-1)d}\right)^{1/(r-1)}$ asymptotically. Furthermore, we prove an analogous low-degree computational threshold of $\left(\frac{\log d}{(r-1)d}\right)^{1/(r-1)}$. Our results recover and generalize recent work of Perkins and the second author on bipartite graphs. While the graph case has been extensively studied, this work is the first to consider statistical-computational gaps of optimization problems on random hypergraphs. Our results suggest that these gaps persist for larger uniformities as well as across many models. A somewhat surprising aspect of the gap for balanced independent sets is that the algorithm achieving the lower bound is a simple degree-1 polynomial.

翻译：我们研究了在具有平均度 $d$ 的 $n$ 个顶点上的 Erdős–Rényi $r$-一致超图中寻找大独立集的算法任务。Krivelevich 和 Sudakov 证明了最大独立集的密度为 $\left(\frac{r\log d}{(r-1)d}\right)^{1/(r-1)}$。我们证明了低次多项式算法类能够找到密度为 $\left(\frac{\log d}{(r-1)d}\right)^{1/(r-1)}$ 的独立集，但无法找到更大的。这扩展并推广了 Gamarnik 和 Sudan、Rahman 和 Virág 以及 Wein 早期在图上的结果，并回答了 Bal 和 Bennett 的一个问题。我们推测该问题的这一统计-计算间隙是成立的。此外，我们通过考察 $r$-部超图来探索这一间隙的普适性。如果存在划分 $V=V_1\cup\cdots\cup V_r$ 使得每条边恰好包含每个集合 $V_i$ 中的一个顶点，则超图 $H=(V,E)$ 是 $r$-部的。我们考虑在每个部分具有 $n$ 个顶点且平均度为 $d$ 的随机 $r$-部超图中寻找大平衡独立集（在每个部分包含相同数量顶点的独立集）的问题。我们证明了最大平衡独立集的渐近密度为 $\left(\frac{r\log d}{(r-1)d}\right)^{1/(r-1)}$。此外，我们证明了一个类似的低度计算阈值 $\left(\frac{\log d}{(r-1)d}\right)^{1/(r-1)}$。我们的结果恢复并推广了 Perkins 和第二作者最近在二分图上的工作。虽然图的情况已被广泛研究，但这项工作首次考虑了随机超图上优化问题的统计-计算间隙。我们的结果表明，这些间隙在更大的一致性以及许多模型中持续存在。平衡独立集间隙的一个有些令人惊讶的方面是，达到下界的算法是一个简单的一次多项式。