The Low-Degree Hardness of Finding Large Independent Sets in Sparse Random Hypergraphs

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(fracrlog d(r-1)dright)1/(r-1). We show that the class of low-degree polynomial algorithms can find independent sets of density left(fraclog d(r-1)dright)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=V1cupcdotscup Vr such that each edge contains exactly one vertex from each set Vi. 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(fracrlog d(r-1)dright)1/(r-1) asymptotically. Furthermore, we prove an analogous low-degree computational threshold of left(fraclog d(r-1)dright)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.