All finite lattices are stable matching lattices
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911574827991040 |
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| author | En, Christopher Faenza, Yuri |
| author_facet | En, Christopher Faenza, Yuri |
| contents | We show that all finite lattices, including non-distributive lattices, arise as stable matching lattices when all agents have path-independent choice functions. This result answers an open question of Blair~\cite{blair1988lattice}. In the process, we introduce new tools to reason on general lattices for optimization purposes: the \emph{partial representation} of a lattice, which partially extends Birkhoff's representation theorem to non-distributive lattices; the \emph{distributive closure} of a lattice, which gives such a partial representation; and \emph{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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_17916 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | All finite lattices are stable matching lattices En, Christopher Faenza, Yuri Discrete Mathematics Theoretical Economics We show that all finite lattices, including non-distributive lattices, arise as stable matching lattices when all agents have path-independent choice functions. This result answers an open question of Blair~\cite{blair1988lattice}. In the process, we introduce new tools to reason on general lattices for optimization purposes: the \emph{partial representation} of a lattice, which partially extends Birkhoff's representation theorem to non-distributive lattices; the \emph{distributive closure} of a lattice, which gives such a partial representation; and \emph{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. |
| title | All finite lattices are stable matching lattices |
| topic | Discrete Mathematics Theoretical Economics |
| url | https://arxiv.org/abs/2504.17916 |