Conditional Hardness Results for Massively Parallel Computation from Distributed Lower Bounds

We present the first conditional hardness results for massively parallel algorithms for some central graph problems including (approximating) maximum matching, vertex cover, maximal independent set, and coloring. In some cases, these hardness results match or get close to the state of the art algorithms. Our hardness results are conditioned on a widely believed conjecture in massively parallel computation about the complexity of the connectivity problem. We also note that it is known that an unconditional variant of such hardness results might be somewhat out of reach for now, as it would lead to considerably improved circuit complexity lower bounds and would concretely imply that NC
₁
is a proper subset of P. We obtain our conditional hardness result via a general method that lifts unconditional lower bounds from the well-studied LOCAL model of distributed computing to the massively parallel computation setting.