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

2026-09-26

AI Summary

This study addresses the challenge of incorporating structural priors, such as degree constraints, into graph reconstruction. To this end, it proposes the CPD-PIFM framework, which introduces a novel intra-sampling primal-dual guidance mechanism. By dynamically evolving Lagrange multipliers along sampling trajectories, this approach guides the flow matching process to satisfy structural constraints without requiring retraining. Theoretically, we prove that the proposed mechanism preserves permutation equivariance and establish an upper bound on the terminal relaxation error. Empirically, CPD-PIFM improves feasibility by 11–26 percentage points across benchmark datasets, outperforming fixed-guidance methods while eliminating the need for constraint-specific hyperparameter tuning.

Abstract

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.

Problem

• Graph Reconstruction
• Flow Matching
• Structural Constraints
• Partial Observations
• Side Information

Innovation

• Flow Matching
• Graph Reconstruction
• Primal-Dual Guidance
• Lagrange Multipliers
• Constrained Sampling