The diffusion time scale, which determines how rapidly a networked system approaches equilibrium, has attracted broad interest. Most studies treat this time scale as the outcome of prescribed changes in network structure or edge weights. Here, we invert this perspective: to accelerate the dynamics to a target time scale, what is the minimum total added weight required, and how should it be allocated? We develop a Time-Weight Matching (TWM) framework that determines the feasible target range and gives an analytical solution for the optimal allocation. This allocation is governed not by edge importance alone, but by a balance between individual edge importance and edge-edge structural correlation. Such a balance determines whether the added weight should be concentrated on a single edge or distributed cooperatively across multiple edges. Furthermore, the framework predicts a weight-reversal phenomenon, where the edge receiving the largest weight switches as the target time scale varies. These results show how desired time scales reshape the structural roles of edges in network dynamics, offering both theoretical insights and practical guidelines for target-driven network design.