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：我们研究了在平均度为$d$的Erdos-Renyi $r$-一致超图中寻找大规模独立集的算法任务，该超图包含$n$个顶点。Krivelevich与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，低度多项式算法能发现的独立集密度上限为$\left(\frac{\log d}{(r-1)d}\right)^{1/(r-1)}$，且无法超越此界限。该结果扩展并推广了Gamarnik与Sudan、Rahman与Virag以及Wein在图上取得的早期成果，并回答了Bal与Bennett提出的问题。我们推测此类统计-计算差距在该问题上普遍存在。此外，我们通过考察$r$-部超图探索了该差距的普适性。若超图$H=(V,E)$满足存在划分$V=V_1\cup\cdots\cup V_r$使得每条边恰包含每个$V_i$中的一个顶点，则称其为$r$-部超图。本文考虑了在随机$r$-部超图中寻找大规模平衡独立集（各分划中顶点数相等的独立集）的问题，其中每个分划包含$n$个顶点且平均度为$d$。我们证明了平衡独立集的最大密度渐近为$\left(\frac{r\log d}{(r-1)d}\right)^{1/(r-1)}$，并进一步建立了其低度计算阈值$\left(\frac{\log d}{(r-1)d}\right)^{1/(r-1)}$。该结果恢复并推广了Perkins与本文第二作者在二分图上的最新工作。尽管图情形已被广泛研究，本文首次考虑了随机超图上优化问题的统计-计算差距。我们的结果表明，这类差距在更高一致性及多种模型中持续存在。关于平衡独立集差距的一个有趣现象在于，达到下界的算法仅为简单的1次多项式。