  We report a novel approach to the question of whether and how the ground
state can be achieved in square artificial spin ices where frustration is
incomplete. We identify two types of disorder: quenched disorder in the island
response to fields and disorder in the sequence of driving fields. Numerical
simulations show that quenched disorder can lead to final states with lower
energy, and disorder in the driving fields always lowers the final energy
attained by the system. We use a network picture to understand these two
effects: disorder in island responses creates new dynamical pathways, and
disorder in driving fields allows more pathways to be followed.
