  The energy landscape for the random-field Ising model (RFIM) is complex, yet
algorithms such as the push-relabel algorithm exist for computing the exact
ground state of an RFIM sample in time polynomial in the sample volume.
Simulations were carried out to investigate the scaling properties of the
push-relabel algorithm. The time evolution of the algorithm was studied along
with the statistics of an auxiliary potential field. At very small random
fields, the algorithm dynamics are closely related to the dynamics of
two-species annihilation, consistent with fractal statistics for the
distribution of minima in the potential (``height''). For $d=1,2$, a
correlation length diverging at zero disorder sets a cutoff scale for the
magnitude of the height field; our results are most consistent with a power-law
correction to the exponential scaling of the correlation length with disorder
in $d=2$. Near the ferromagnetic-paramagnetic transition in $d=3$, the time to
find a solution diverges with a dynamic critical exponent of $z=0.93\pm0.06$
for a priority queue version and $z=0.43\pm0.06$ for a first-in first-out queue
version of the algorithm. The links between the evolution of auxiliary fields
in algorithmic time and the static physical properties of the RFIM ground state
provide insight into the physics of the RFIM and a better understanding of how
the algorithm functions.
