Non-distributive Lattices, Stable Matchings, and Linear Optimization

2025-04-24

AI Summary

This paper investigates the expressive power of stable matching lattices under standard choice functions and the computational complexity of computing minimum-cost stable matchings. Addressing two open questions—(1) whether every finite lattice (including non-distributive ones) can be realized as a stable matching lattice, and (2) whether minimum-cost stable matching is NP-hard—we introduce novel techniques based on partial lattice representation, distributive closure, and join constraints. Our approach extends stable matching lattice theory to arbitrary finite lattices for the first time. We construct a computationally tractable representation framework for non-distributive lattices, circumventing the distributivity requirement of Birkhoff’s representation theorem. Moreover, we establish, under standard assumptions, that computing a minimum-cost stable matching is NP-hard. These results forge a deep connection between lattice structure and matching optimization, yielding a new structural and algorithmic paradigm for stable matching theory.

Technology Category

• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

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.

Problem

• Studying stable matching lattices in non-distributive finite lattices
• Introducing tools for lattice optimization and partial representation
• Proving NP-hardness of minimum cost stable matching problem

Innovation

• Partial representation extends Birkhoff's theorem
• Distributive closure enables partial representation
• Join constraints complete lattice representation