Beyond Reachability: When Edit Order Preserves Couplings in Graph Flow Matching
Zilin Li, Zixing Song
Shortest-path optimality does not guarantee that a memoryless graph generator preserves a prescribed source-target correspondence. We study whether reordering legal shortest edits can retain this correspondence without changing the pairing, mandatory edits, or sampling budget. Under a common Poisson clock, we establish an exact criterion for endpoint coupling recoverability: the source-blind Markov projection attains the unavoidable clock-error floor if and only if routes to distinct targets do not share graph states. Routes to the same target may merge without sacrificing endpoint fidelity. This characterization leads to an exact finite-state routing formulation and a legality-preserving scheduling procedure with recovery certificates. A marker-free construction demonstrates how shared edge structure can retain source distinctions, while a capacity-constrained family exhibits a sharp separation in paired error between edit orders with identical marginal error. Finally, a target-conditioned aliasing bound connects structural route design to the terminal fidelity of a learned graph-only generator. These results characterize when legal edit ordering can preserve a prescribed coupling under fixed operational costs.
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