The Mpemba effect refers to the counterintuitive situation in which a system initially farther
from equilibrium can relax faster than one that starts closer to it. In quantum systems,
the effect is enriched by the presence of coherent dynamics, dissipation, and metastable
manifolds associated with long-lived Liouvillian modes. Here we demonstrate a giant
Mpemba effect in open quantum systems, where relaxation can be either hyper-accelerated
or dramatically slowed depending on the initial state. We focus on weakly-coupled particleconserving
bosonic networks, each of which independently relaxes rapidly to a unique
stationary state. When a weak coherent interaction is introduced, the composite system
typically develops slow metastable modes and a hierarchy of relaxation timescales. We
show that by tailoring the interaction Hamiltonian, these slow modes can be effectively
suppressed for a broad class of initial states satisfying a minimal global requirement,
enabling ultrafast relaxation even when the system starts far from equilibrium. Conversely,
other initial states—sometimes arbitrarily close to the stationary state—may remain trapped
in the metastable manifold and decay anomalously slowly. This mechanism provides a
general route to engineer giant Mpemba effects, offering new possibilities for controlling
dissipative dynamics, accelerating state preparation, and manipulating relaxation processes
in complex quantum devices.