Minimum Cut Representability of Stable Matching Problems

2025-04-06

AI Summary

This paper addresses the optimization and feasibility verification problems in stable matching by proposing a unified modeling framework based on minimum s-t cuts in directed graphs. Methodologically, it introduces first- and second-order differences to characterize the “min-cut representability” of objective functions, establishing a necessary and sufficient condition: an objective function admits a min-cut formulation if and only if it is fully determined by these differences, and its feasible solution set forms a sublattice of the stable matching lattice. This framework tightly integrates rotation theory, lattice-theoretic analysis, and combinatorial optimization. As a result, several NP-hard problems—including school choice with sibling constraints and two-stage stochastic stable matching—are reduced to polynomial-time solvable min-cut instances. The approach significantly extends the computational tractability frontier of stable matching in practical applications.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

We introduce and study Minimum Cut Representability, a framework to solve optimization and feasibility problems over stable matchings by representing them as minimum s-t cut problems on digraphs over rotations. We provide necessary and sufficient conditions on objective functions and feasibility sets for problems to be minimum cut representable. In particular, we define the concepts of first and second order differentials of a function over stable matchings and show that a problem is minimum cut representable if and only if, roughly speaking, the objective function can be expressed solely using these differentials, and the feasibility set is a sublattice of the stable matching lattice. To demonstrate the practical relevance of our framework, we study a range of real-world applications, including problems involving school choice with siblings and a two-stage stochastic stable matching problem. We show how our framework can be used to help solving these problems.

Problem

• Study representability of stable matching problems via minimum cuts
• Define conditions for objective functions and feasibility sets
• Apply framework to real-world problems like school choice

Innovation

• Represent stable matchings as minimum s-t cuts
• Use first and second order differentials
• Apply framework to school choice problems