A Breezing Proof of the KMW Bound

In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW) proved a hardness result for several fundamental graph problems in the LOCAL model: For any (randomized) algorithm, there are graphs with n nodes and maximum degree ∆ on which Ω(min{plog n/log log n, log ∆/log log ∆}) (expected) communication rounds are required to obtain polylogarithmic approximations to a minimum vertex cover, minimum dominating set, or maximum matching. Via reduction, this hardness extends to symmetry breaking tasks like finding maximal independent sets or maximal matchings. Today, more than 15 years later, there is still no proof of this result that is easy on the reader. Setting out to change this, in this work, we provide a simplified proof of the main step in showing the KMW lower bound. Our key argument is algorithmic, and it relies on an invariant that can be readily verified from the generation rules of the lower bound graphs.