  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 input graphs with $n$ nodes
and maximum degree $\Delta$ on which $\Omega(\min\{\sqrt{\log n/\log \log
n},\log \Delta/\log \log \Delta\})$ (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 fully self-contained and $\mathit{simple}$ proof of the KMW
lower bound. The key argument is algorithmic, and it relies on an invariant
that can be readily verified from the generation rules of the lower bound
graphs.
