Topology classes supporting minimum-weight perfect matching lower bounds

Characterize a large class of graph topologies for which differentially private minimum-weight perfect matching under the \(\ell_1\) edge-weight neighboring relation necessarily incurs strong additive error lower bounds, and determine the specific graph property that such a class should satisfy.

Background

The paper establishes tight worst-case lower bounds for private minimum-weight perfect matching by encoding data into dense graph topologies. In contrast, its topology-dependent results for broader graph families are given for minimum spanning trees and hierarchical clustering, not for minimum-weight perfect matching.

The authors explicitly leave unresolved whether analogous lower bounds can be proved for a large class of minimum-weight perfect matching topologies and, if so, which structural property—such as expansion, connectivity, or another topology parameter—should define that class.

References

For MWPM, it remains open whether lower bounds can be derived for a large class of topologies, and what specific property that class should satisfy.

— Lower Bounds for Private Graph Optimization Problems using Reconstruction Attacks  (2609.10877 - Imola et al., 9 Sep 2026) in Section 7, “Conclusion and Future Work”