  The average time required for an open quantum system to reach a steady state
(the steady-state time) is generally determined through a competition of
coherent and incoherent (dissipative) dynamics. Here, we study this competition
for a ubiquitous central-spin system, corresponding to a `central' spin-1/2
coherently coupled to ancilla spins and undergoing dissipative spin relaxation.
The ancilla system can describe $N$ spins-1/2 or, equivalently, a single large
spin of length $I=N/2$. We find exact analytical expressions for the
steady-state time in terms of the dissipation rate, resulting in a minimal
(optimal) steady-state time at an optimal value of the dissipation rate,
according to a universal curve. Due to a collective-enhancement effect, the
optimized steady-state time grows only logarithmically with increasing $N=2I$,
demonstrating that the system size can be grown substantially with only a
moderate cost in steady-state time. This work has direct applications to the
rapid initialization of spin qubits in quantum dots or bound to donor
impurities, to dynamic nuclear-spin polarization protocols, and may provide key
intuition for the benefits of error-correction protocols in quantum annealing.
