The Popular Dimension of Matchings
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arXiv
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| Main Authors: | , , , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916977163894784 |
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| author | Connor, Frank Langevin, Louis-Roy Ndiaye, Ndiamé Totschnig, Agnès Vasishta, Rohit Vetta, Adrian |
| author_facet | Connor, Frank Langevin, Louis-Roy Ndiaye, Ndiamé Totschnig, Agnès Vasishta, Rohit Vetta, Adrian |
| contents | We study popular matchings in three classical settings: the house allocation problem, the marriage problem, and the roommates problem. In the popular matching problem, (a subset of) the vertices in a graph have preference orderings over their potential matches. A matching is popular if it gets a plurality of votes in a pairwise election against any other matching. Unfortunately, popular matchings typically do not exist. So we study a natural relaxation, namely popular winning sets which are a set of matchings that collectively get a plurality of votes in a pairwise election against any other matching. The $\textit{popular dimension}$ is the minimum cardinality of a popular winning set, in the worst case over the problem class.
We prove that the popular dimension is exactly $2$ in the house allocation problem, even if the voters are weighted and ties are allowed in their preference lists. For the marriage problem and the roommates problem, we prove that the popular dimension is between $2$ and $3$, when the agents are weighted and/or their preferences orderings allow ties. In the special case where the agents are unweighted and have strict preference orderings, the popular dimension of the marriage problem is known to be exactly $1$ and we prove the popular dimension of the roommates problem is exactly $2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_25150 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Popular Dimension of Matchings Connor, Frank Langevin, Louis-Roy Ndiaye, Ndiamé Totschnig, Agnès Vasishta, Rohit Vetta, Adrian Computer Science and Game Theory Discrete Mathematics Data Structures and Algorithms Combinatorics We study popular matchings in three classical settings: the house allocation problem, the marriage problem, and the roommates problem. In the popular matching problem, (a subset of) the vertices in a graph have preference orderings over their potential matches. A matching is popular if it gets a plurality of votes in a pairwise election against any other matching. Unfortunately, popular matchings typically do not exist. So we study a natural relaxation, namely popular winning sets which are a set of matchings that collectively get a plurality of votes in a pairwise election against any other matching. The $\textit{popular dimension}$ is the minimum cardinality of a popular winning set, in the worst case over the problem class. We prove that the popular dimension is exactly $2$ in the house allocation problem, even if the voters are weighted and ties are allowed in their preference lists. For the marriage problem and the roommates problem, we prove that the popular dimension is between $2$ and $3$, when the agents are weighted and/or their preferences orderings allow ties. In the special case where the agents are unweighted and have strict preference orderings, the popular dimension of the marriage problem is known to be exactly $1$ and we prove the popular dimension of the roommates problem is exactly $2$. |
| title | The Popular Dimension of Matchings |
| topic | Computer Science and Game Theory Discrete Mathematics Data Structures and Algorithms Combinatorics |
| url | https://arxiv.org/abs/2509.25150 |