  We consider the Ising model on a dense Erd\H{o}s--R\'enyi random graph,
$\mathcal G(N,p)$, with $p>0$ fixed---equivalently, a disordered Curie--Weiss
Ising model with $\mbox{Ber}(p)$ couplings---at zero temperature. The disorder
may induce local energy minima in addition to the two uniform ground states. In
this paper we prove that, starting from a typical initial configuration, the
zero-temperature dynamics avoids all such local minima and absorbs into a
predetermined one of the two uniform ground states. We relate this to the local
MINCUT problem on dense random graphs; namely with high probability, the greedy
search for a local MINCUT of $\mathcal G(N,p)$ with $p>0$ fixed, started from a
uniform random partition, fails to find a non-trivial cut. In contrast, in the
disordered Curie--Weiss model with heavy-tailed couplings, we demonstrate that
zero-temperature dynamics has positive probability of absorbing in a random
local minimum different from the two homogenous ground states.
