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arXiv · 2606.10133

k-Anonymity by Partitions Maximizes Perfect Matchings

Ewa J. Infeld
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Abstract

The number of perfect matchings in a user-behavior bipartite graph is a natural measure of anonymity: more matchings mean greater uncertainty for an attacker. A fundamental question is which graph structure maximizes this count for a fixed infrastructure cost, represented by the number of edges. We prove that the answer is $k$-anonymity by partitions. Using Brègman's Theorem, we show that partitioning users into equal-sized groups and making each group a complete bipartite component achieves the theoretical upper bound on perfect matchings. For edge counts where an exact partition is impossible, we construct a family of graphs that asymptotically attains this bound as the group size grows. We further prove that this optimality is robust: after an attacker de-anonymizes a user by the most damaging choice, the resulting graph is still a partition graph and remains optimal. Together, these results provide a combinatorial justification for the widespread use of $k$-anonymity by partitions in anonymity system design.

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Ewa J. Infeld. 2026-06-08. k-Anonymity by Partitions Maximizes Perfect Matchings. https://arxiv.org/abs/2606.10133

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