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A new approach to bipartite stable matching optimization

Tamás Fleiner, András Frank, Tamás Király

TL;DR

This work develops two complementary frameworks for optimization over bipartite stable matchings. The ring-set/digraph approach encodes stable matchings as stable $s^*$-$t^*$-cuts in a compact digraph, yielding strong min-max formulas and strongly polynomial algorithms for packing and covering problems, including computing $\ell$ disjoint cheapest matchings and the minimum number of matchings covering all stable edges. The second, poset-based view constructs a $G$-induced poset on stable edges whose maximal antichains correspond to stable matchings, enabling Dilworth–Mirsky–Greene–Kleitman results and efficient algorithms for level-fair and related fairness notions. Together, these contributions unify optimization over stable matchings with network-flow and poset theories, providing practical polynomial-time methods and new structural insights with potential extensions to matroid kernels and related combinatorial objects.

Abstract

As a common generalization of previously solved optimization problems concerning bipartite stable matchings, we describe a strongly polynomial network flow based algorithm for computing $\ell$ disjoint stable matchings with minimum total cost. The major observation behind the approach is that stable matchings, as edge sets, can be represented as certain cuts of an associated directed graph. This allows us to use results on disjoint cuts directly to answer questions about disjoint stable matchings. We also provide a construction that represents stable matchings as maximum-size antichains in a partially ordered set (poset), which enables us to apply the theorems of Dilworth, Mirsky, Greene and Kleitman directly to stable matchings. Another consequence of these approaches is a min-max formula for the minimum number of stable matchings covering all stable edges.

A new approach to bipartite stable matching optimization

TL;DR

This work develops two complementary frameworks for optimization over bipartite stable matchings. The ring-set/digraph approach encodes stable matchings as stable --cuts in a compact digraph, yielding strong min-max formulas and strongly polynomial algorithms for packing and covering problems, including computing disjoint cheapest matchings and the minimum number of matchings covering all stable edges. The second, poset-based view constructs a -induced poset on stable edges whose maximal antichains correspond to stable matchings, enabling Dilworth–Mirsky–Greene–Kleitman results and efficient algorithms for level-fair and related fairness notions. Together, these contributions unify optimization over stable matchings with network-flow and poset theories, providing practical polynomial-time methods and new structural insights with potential extensions to matroid kernels and related combinatorial objects.

Abstract

As a common generalization of previously solved optimization problems concerning bipartite stable matchings, we describe a strongly polynomial network flow based algorithm for computing disjoint stable matchings with minimum total cost. The major observation behind the approach is that stable matchings, as edge sets, can be represented as certain cuts of an associated directed graph. This allows us to use results on disjoint cuts directly to answer questions about disjoint stable matchings. We also provide a construction that represents stable matchings as maximum-size antichains in a partially ordered set (poset), which enables us to apply the theorems of Dilworth, Mirsky, Greene and Kleitman directly to stable matchings. Another consequence of these approaches is a min-max formula for the minimum number of stable matchings covering all stable edges.
Paper Structure (31 sections, 34 theorems, 50 equations)

This paper contains 31 sections, 34 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Notions and notation
  3. Stable matchings
  4. Optimization over ring-sets defined on a digraph
  5. Largest $h$-independent packing of ring-set members
  6. A two-phase greedy algorithm for the special case $h\equiv 1$
  7. Minimizing the out-capacity $\delta _g(Z)$ over the members of a ring-set
  8. Associating a digraph with the preference system on $G$
  9. Stable matchings of $G$ versus stable $s\sp *t\sp *$-cuts of $D$
  10. Cheapest stable matchings
  11. How to find a cheapest stable matching
  12. Multiple cost-function
  13. Forbidden and forced edges
  14. Packing cheapest stable matchings
  15. Fair stable matchings
  16. ...and 16 more sections

Key Result

THEOREM 2.1

For digraph $D=(V,A)$, function $h$, and ring-set ${\cal R}_0$, we have $\nu _h = \tau _h$. By a single application of Dijkstra's shortest path algorithm, both a $h$-independent family of $\nu _h$ (not necessarily distinct) members of ${\cal R}_0$ and a subset $L\subseteq A$ of $\tau _h$ ($=\nu _h)$

Theorems & Definitions (53)

  • Claim 1.1
  • Claim 1.2
  • Claim 1.3
  • THEOREM 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Claim 3.1
  • Claim 3.2
  • Claim 3.3
  • ...and 43 more