Understanding Popular Matchings via Stable Matchings

@article{Cseh2022UnderstandingPM,
  title={Understanding Popular Matchings via Stable Matchings},
  author={{\'A}gnes Cseh and Yuri Faenza and Telikepalli Kavitha and Vladlena Powers},
  journal={SIAM J. Discret. Math.},
  year={2022},
  volume={36},
  pages={188-213}
}
An instance of the marriage problem is given by a graph G together with, for each vertex of G, a strict preference order over its neighbors. A matching M of Gis popular in the marriage instance if M does not lose a head-to-head election against anymatching where vertices are voters. Every stable matching is a min-size popular matching; another subclass of popular matchings that always exist and can be easily computed is theset of dominant matchings. A popular matching M is dominant if M wins… 

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References

SHOWING 1-10 OF 35 REFERENCES

Popular matchings in the stable marriage problem

Popularity, Mixed Matchings, and Self-duality

TLDR
The popular fractional matching polytope PG is half-integral and in the special case where a stable matching in G is a perfect matching, thispolytope is integral, which implies that there is always a max-utility popular mixed matching Π, and this result carries over to the roommates problem, where the graph G need not be bipartite.

Popular Half-Integral Matchings

TLDR
It is shown that every popular half-integral matching is equivalent to a stable matching in a larger graph G^*.

An Efficient Algorithm for the "Stable Roommates" Problem

Popular Matchings and Limits to Tractability

TLDR
It is NP-complete to decide if $G$ admits a popular matching that is neither stable nor dominant, and a number of related hardness results, such as (tight) inapproximability of the maximum weight popular matching problem.

Quasi-popular Matchings, Optimality, and Extended Formulations

TLDR
This version of the paper goes beyond the conference version in the following two points: (i) the algorithm for finding a quasi-popular matching of cost at most that of a min-cost popular fractional matching is new; (ii) the proofs from Section 6.3 are now self-contained.

An efficient algorithm for the “optimal” stable marriage

TLDR
By exploiting the structure of the set of all stable matchings, and using graph-theoretic methods, an O(n4) algorithm for this problem is derived and achieves the objective of maximizing the average “satisfaction” of all people.

A Size-Popularity Tradeoff in the Stable Marriage Problem

  • T. Kavitha
  • Mathematics, Computer Science
    SIAM J. Comput.
  • 2014
TLDR
The first linear time algorithm for computing a maximum size popular matching in G, a bipartite graph where each vertex ranks its neighbors in a strict order of preference, is shown.

Popular Matchings in the Marriage and Roommates Problems

TLDR
This paper investigates the relationship between popularity and stability, and describes efficient algorithms to test a matching for popularity in these settings, including its special (bipartite) case, the Marriage Problem.

Popular mixed matchings

TLDR
It is shown that popular mixed matchings always exist and polynomial time algorithms for finding them are designed and they are studied to give tight bounds on the price of anarchy and price of stability of the popular matching problem.