Feasible Flow Matching for Graph Reconstruction via Within-Sampling Primal-Dual Guidance

Graph reconstruction from partial observations often comes with structural side information, such as degree bounds, triangle counts, or an edge-density band. Prior-Informed Flow Matching (PIFM) reconstructs graphs by transporting a local prior toward the graph distribution, but it provides no mechanism to incorporate this side information. We put forth Constrained Primal-Dual PIFM (CPD-PIFM), which augments the sampler with Lagrange multipliers that evolve along each trajectory. The multipliers respond to constraint violations at a predicted endpoint and guide subsequent sampling steps without retraining. We prove that the sampler inherits PIFM's permutation equivariance and bound its expected terminal slack by a term that decays as the inverse square root of the number of steps, plus two approximation terms. On three link-prediction benchmarks and nine combinations of datasets and constraints, CPD-PIFM raises feasibility by 11-26 percentage points and remains competitive with fixed guidance without selecting a separate multiplier for each constraint.