Subdividing non-conforming T-mesh layouts into conforming quadrangular meshes is a core component of state-of-the-art (re-)meshing methods. Typically, the required constrained assignment of integer lengths to T-Mesh edges is left to generic branch-and-cut solvers, greedy heuristics, or a combination of the two. This either does not scale well with input complexity or delivers suboptimal result quality. We introduce the Minimum-Deviation-Flow Problem in bi-directed networks (Bi-MDF) and demonstrate its use in modeling and efficiently solving a variety of T-Mesh quantization problems. We develop a fast approximate solver as well as an iterative refinement algorithm based on graph matching that solves Bi-MDF exactly. Compared to the state-of-the-art QuadWild implementation on the authors' 300 dataset, our exact solver finishes after only 0.49% (total 17.06s) of their runtime (3491s) and achieves 11% lower energy while an approximation is computed after 0.09% (3.19s) of their runtime at the cost of 24% increased energy. A novel half-arc-based T-Mesh quantization formulation extends the feasible solution space to include previously unattainable quad meshes. The Bi-MDF problem is more general than our application in layout quantization, potentially enabling similar speedups for other optimization problems that fit into the scheme, such as quad mesh refinement.