Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Stable matchings, choice functions, and linear orders

Alexander V. Karzanov

TL;DR

The paper analyzes stable matchings in a bipartite graph where workers have linear preferences and firms use choice functions satisfying consistency, substitutability, and cardinal monotonicity (CBM). It develops a combinatorial description of rotations, constructs the rotation poset in O(|E|^2) time, and proves a lattice isomorphism between stable matchings and ideals of the poset, yielding a compact affine representation of stable matchings via an order-polytope framework. The work extends to arbitrary quotas on workers, preserves key properties, and shows how to solve minimum-cost stability problems by reducing to a minimum cut in a derived graph. It also connects CBM to the sequential-choice (S-model) by a reduction that preserves the lattice structure, enabling efficient transfer of algorithms and representations across models. These results provide both structural insight and practical algorithms for stability analysis and optimization in generalized bipartite markets.

Abstract

We consider a model of stable edge sets (``matchings'') in a bipartite graph $G=(V,E)$ in which the preferences for vertices of one side (``firms'') are given via choice functions subject to standard axioms of consistency, substitutability and cardinal monotonicity, whereas the preferences for the vertices of the other side (``workers'') via linear orders. For such a model, we present a combinatorial description of the structure of rotations and develop an algorithm to construct the poset of rotations, in time $O(|E|^2)$ (including oracle calls). As consequences, one can obtain a ``compact'' affine representation of stable matchings and efficiently solve some related problems. Keywords: bipartite graph, choice function, linear preferences, stable matching, affine representation, sequential choice

Stable matchings, choice functions, and linear orders

TL;DR

The paper analyzes stable matchings in a bipartite graph where workers have linear preferences and firms use choice functions satisfying consistency, substitutability, and cardinal monotonicity (CBM). It develops a combinatorial description of rotations, constructs the rotation poset in O(|E|^2) time, and proves a lattice isomorphism between stable matchings and ideals of the poset, yielding a compact affine representation of stable matchings via an order-polytope framework. The work extends to arbitrary quotas on workers, preserves key properties, and shows how to solve minimum-cost stability problems by reducing to a minimum cut in a derived graph. It also connects CBM to the sequential-choice (S-model) by a reduction that preserves the lattice structure, enabling efficient transfer of algorithms and representations across models. These results provide both structural insight and practical algorithms for stability analysis and optimization in generalized bipartite markets.

Abstract

We consider a model of stable edge sets (``matchings'') in a bipartite graph in which the preferences for vertices of one side (``firms'') are given via choice functions subject to standard axioms of consistency, substitutability and cardinal monotonicity, whereas the preferences for the vertices of the other side (``workers'') via linear orders. For such a model, we present a combinatorial description of the structure of rotations and develop an algorithm to construct the poset of rotations, in time (including oracle calls). As consequences, one can obtain a ``compact'' affine representation of stable matchings and efficiently solve some related problems. Keywords: bipartite graph, choice function, linear preferences, stable matching, affine representation, sequential choice
Paper Structure (12 sections, 11 theorems, 15 equations)

This paper contains 12 sections, 11 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Basic definitions and settings
  3. Active graph and rotations
  4. Active graph.
  5. Rotations.
  6. Poset of rotations
  7. Constructions
  8. Finding the generating graph for $({\cal R},\lessdot)$.
  9. Finding the initial stable matching $X^{\rm min}$.
  10. Affine representation and stable matchings of minimum cost
  11. Arbitrary quotas on $W$
  12. Model of stability with sequential choices

Key Result

Proposition 3.1

For each $L\in{\cal L}(X)$, the matching $X':=(X-L^-)\cup L^+$ is stable and satisfies $X'\succ X$.

Theorems & Definitions (11)

  • Proposition 3.1
  • Proposition 3.2
  • Lemma 3.3
  • Lemma 4.1
  • Proposition 4.2
  • Lemma 5.1
  • Proposition 5.2
  • Proposition 5.3
  • Theorem 5.4
  • Corollary 6.1
  • ...and 1 more