No Truly Sublogarithmic-Round Algorithm for Distributed MIS in the LOCAL Model

We prove that $o(\frac{\log n}{\log\log n})$-round randomized LOCAL algorithms find a maximal independent set (MIS) with subconstant probability on some input. This shows that the 40-year-old $O(\log n)$-round randomized CONGEST algorithm of Luby and Alon, Babai, and Itai is nearly optimal. Our lower bound improves upon the $Ω(\sqrt{\log n})$-round lower bound of Khoury and Schild and nearly closes the 20+ year-old gap between the $O(\log n)$-round algorithm and the $Ω(\sqrt{\frac{\log n}{\log\log n}})$-round lower bound of Kuhn, Moscibroda, and Wattenhofer. To close this gap, one must show a lower bound for a graph that is not locally treelike. We do this by starting with a $D$-regular, $Θ(\log_D n)$-girth graph, with $D = Θ(\log n)$. Crucially, this is not our final graph. We obtain our hard instance by \emph{replacing} each vertex of this treelike graph with a clique on $n^c$ vertices for some constant $c \in (0,1)$. We then replace each edge with a bipartite Erdős--Rényi graph between the corresponding cliques, with each edge present independently with probability $1/2$. We analyze arbitrary LOCAL algorithms on our non-treelike hard instance with a novel round elimination approach. We describe algorithms that output a set of vertices in this graph using two metrics: their \emph{occupancy probability} and their \emph{conflict probability}. Our round elimination framework takes an $r$-round algorithm and produces an $(r-1)$-round algorithm where occupancy and conflict probabilities collectively slightly degrade, but individually can significantly improve or degrade. This round elimination result is obtained by exploiting the lack of large biclique complements in dense bipartite Erdős--Rényi graphs, which in turn allows us to bound the entropy of the output distribution for any independent set algorithm.