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We present a general framework for modifying quantum approximate optimization algorithms (QAOA) to solve constrained network flow problems. By exploiting an analogy between flow constraints and Gauss's law for electromagnetism, we design lattice quantum electrodynamics (QED) inspired mixing Hamiltonians that preserve flow constraints throughout the QAOA process. This results in an exponential reduction in the size of the configuration space that needs to be explored, which we show through numerical simulations, yields higher quality approximate solutions compared to the original QAOA routine. We outline a specific implementation for edge-disjoint path (EDP) problems related to traffic congestion minimization, numerically analyze the effect of initial state choice, and explore trade-offs between circuit complexity and qubit resources via a particle-vortex duality mapping. Comparing the effect of initial states reveals that starting with an ergodic (unbiased) superposition of solutions yields better performance than beginning with the mixer ground-state, suggesting a departure from the "short-cut to adiabaticity" mechanism often used to motivate QAOA.