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Non-distributive Lattices, Stable Matchings, and Linear Optimization

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  • Christopher En
  • Yuri Faenza

Abstract

We show that all finite lattices, including non-distributive lattices, arise as stable matching lattices under standard assumptions on choice functions. In the process, we introduce new tools to reason on general lattices for optimization purposes: the partial representation of a lattice, which partially extends Birkhoff's representation theorem to non-distributive lattices; the distributive closure of a lattice, which gives such a partial representation; and join constraints, which can be added to the distributive closure to obtain a representation for the original lattice. Then, we use these techniques to show that the minimum cost stable matching problem under the same standard assumptions on choice functions is NP-hard, by establishing a connection with antimatroid theory.

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  • Christopher En & Yuri Faenza, 2025. "Non-distributive Lattices, Stable Matchings, and Linear Optimization," Papers 2504.17916, arXiv.org.
    • Handle: RePEc:arx:papers:2504.17916
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    References listed on IDEAS

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