Maximum-utility Popular Matchings with Bounded Instability

Titelangaben

Schlotter, Ildiko ; Cseh, Ágnes:
Maximum-utility Popular Matchings with Bounded Instability.
In: ACM Transactions on Computation Theory. Bd. 17 (8 März 2025) Heft 1 . - 6.
ISSN 1942-3462
DOI: https://doi.org/10.1145/3711843

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Abstract

In a graph where vertices have preferences over their neighbors, a matching is called popular if it does not lose a head-to-head election against any other matching when the vertices vote between the matchings. Popular matchings can be seen as an intermediate category between stable matchings and maximum-size matchings. In this article, we aim to maximize the utility of a matching that is popular but admits only a few blocking edges.We observe that, for general graphs, finding a popular matching with at most one blocking edge is already NP-complete. For bipartite instances, we study the problem of finding a maximum-utility popular matching with a bound on the number (or, more generally, the cost) of blocking edges applying a multivariate approach. We show classical and parameterized hardness results for severely restricted instances. By contrast, we design an algorithm for instances where preferences on one side admit a master list and show that this algorithm is roughly optimal.