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Popular Maximum-Utility Matchings with Matroid Constraints

Gergely Csáji, Tamás Király, Kenjiro Takazawa, Yu Yokoi

TL;DR

The paper advances the theory of popular matchings by incorporating matroid constraints and, for weights, extends to $M^{\natural}$-concave utilities, yielding polynomial-time algorithms for both one- and two-sided models. The one-sided approach reduces to popular common base problems via matroid-intersection duality and chain structures, while the two-sided model uses matroid kernels and two transformative steps (chains and duplication) to enable a polynomial-time algorithm for a popular critical variant. Together these results generalize prior tractable regimes (including stable and popular bases) to broad matroidal and discrete-concavity settings, enabling practical computation in allocation and matching problems with diversity and capacity constraints. The paper also delineates hardness for near-optimal variants, highlighting fundamental limits and guiding future research on approximate or restricted models in matroid-constrained popular settings.

Abstract

We investigate weighted settings of popular matching problems with matroid constraints. The concept of popularity was originally defined for matchings in bipartite graphs, where vertices have preferences over the incident edges. There are two standard models depending on whether vertices on one or both sides have preferences. A matching $M$ is popular if it does not lose a head-to-head election against any other matching. In our generalized models, one or both sides have matroid constraints, and a weight function is defined on the ground set. Our objective is to find a popular optimal matching, i.e., a maximum-weight matching that is popular among all maximum-weight matchings satisfying the matroid constraints. For both one- and two-sided preferences models, we provide efficient algorithms to find such solutions, combining algorithms for unweighted models with fundamental techniques from combinatorial optimization. The algorithm for the one-sided preferences model is further extended to a model where the weight function is generalized to an M$^\natural$-concave utility function. Finally, we complement these tractability results by providing hardness results for the problems of finding a popular near-optimal matching. These hardness results hold even without matroid constraints and with very restricted weight functions.

Popular Maximum-Utility Matchings with Matroid Constraints

TL;DR

The paper advances the theory of popular matchings by incorporating matroid constraints and, for weights, extends to -concave utilities, yielding polynomial-time algorithms for both one- and two-sided models. The one-sided approach reduces to popular common base problems via matroid-intersection duality and chain structures, while the two-sided model uses matroid kernels and two transformative steps (chains and duplication) to enable a polynomial-time algorithm for a popular critical variant. Together these results generalize prior tractable regimes (including stable and popular bases) to broad matroidal and discrete-concavity settings, enabling practical computation in allocation and matching problems with diversity and capacity constraints. The paper also delineates hardness for near-optimal variants, highlighting fundamental limits and guiding future research on approximate or restricted models in matroid-constrained popular settings.

Abstract

We investigate weighted settings of popular matching problems with matroid constraints. The concept of popularity was originally defined for matchings in bipartite graphs, where vertices have preferences over the incident edges. There are two standard models depending on whether vertices on one or both sides have preferences. A matching is popular if it does not lose a head-to-head election against any other matching. In our generalized models, one or both sides have matroid constraints, and a weight function is defined on the ground set. Our objective is to find a popular optimal matching, i.e., a maximum-weight matching that is popular among all maximum-weight matchings satisfying the matroid constraints. For both one- and two-sided preferences models, we provide efficient algorithms to find such solutions, combining algorithms for unweighted models with fundamental techniques from combinatorial optimization. The algorithm for the one-sided preferences model is further extended to a model where the weight function is generalized to an M-concave utility function. Finally, we complement these tractability results by providing hardness results for the problems of finding a popular near-optimal matching. These hardness results hold even without matroid constraints and with very restricted weight functions.
Paper Structure (21 sections, 19 theorems, 31 equations, 1 figure, 2 tables)

This paper contains 21 sections, 19 theorems, 31 equations, 1 figure, 2 tables.

Table of Contents

  1. Introduction
  2. Previous Models
  3. Our Contributions
  4. Related Works
  5. One-sided Preferences.
  6. Two-sided Preferences.
  7. Matroids and M$^\natural$-concave Functions
  8. One-sided Preferences Models
  9. Finding a Popular Maximum-Weight Common Independent Set
  10. Finding a Popular Maximum-Utility Common Independent Set
  11. Two-sided Preferences Model
  12. Popularity in the Two-sided Preferences Model
  13. Matroid Kernels.
  14. Reducing to the Popular Critical Common Independent Set Problem
  15. Popular Critical Common Independent Set Algorithm
  16. ...and 6 more sections

Key Result

Theorem 1

Given a $1$-partition matroid $M_1=(S, \mathcal{I}_1)$ associated with partial orders $\{\succ_i\}_{i\in [n]}$, an arbitrary matroid $M_2=(S, \mathcal{I}_2)$, and a weight function $w\colon S \to \mathbb{R}$, one can determine the existence of a popular maximum-weight common independent set and find

Figures (1)

  • Figure 1: An instance in Lemma \ref{['lem:cyclelemma']} for the case $\ell=2$ and $K=2$. White and black colors represent the bipartition of the vertex set. A small arrow leaving from each vertex represents the edge that vertex most prefers. Four vertices $v_1, v_2, v_3, v_4$ are corner vertices and all other vertices are connector vertices.

Theorems & Definitions (58)

  • Definition
  • Theorem 1: Tractability in the weighted model with one-sided preferences
  • Definition
  • Theorem 2: Tractability in the M$^\natural$-concave model with one-sided preferences
  • Theorem 3: Tractability in the weighted model with two-sided preferences
  • Definition
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • proof
  • ...and 48 more