Robust Popular Matchings

We study popularity for matchings under preferences. This solution concept captures matchings that do not lose against any other matching in a majority vote by the agents. A popular matching is said to be robust if it is popular among multiple instances. We present a polynomial-time algorithm for deciding whether there exists a robust popular matching if instances only differ with respect to the preferences of a single agent while obtaining NP-completeness if two instances differ only by a downward shift of one alternative by four agents. Moreover, we find a complexity dichotomy based on preference completeness for the case where instances differ by making some options unavailable.Comment: Appears in: Proceedings of the 23rd International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2024

A Structural and Algorithmic Study of Stable Matching Lattices of Multiple Instances

Recently MV18a identified and initiated work on the new problem of understanding structural relationships between the lattices of solutions of two ``nearby'' instances of stable matching. They also gave an application of their work to finding a robust stable matching. However, the types of changes they allowed in going from instance $A$ to $B$ were very restricted, namely, any one agent executes an upward shift. In this paper, we allow any one agent to permute its preference list arbitrarily. Let $M_A$ and $M_B$ be the sets of stable matchings of the resulting pair of instances $A$ and $B$, and let $\mathcal{L}_A$ and $\mathcal{L}_B$ be the corresponding lattices of stable matchings. We prove that the matchings in $M_A \cap M_B$ form a sublattice of both $\mathcal{L}_A$ and $\mathcal{L}_B$ and those in $M_A \setminus M_B$ form a join semi-sublattice of $\mathcal{L}_A$. These properties enable us to obtain a polynomial time algorithm for not only finding a stable matching in $M_A \cap M_B$, but also for obtaining the partial order, as promised by Birkhoff's Representation Theorem, thereby enabling us to generate all matchings in this sublattice. Our algorithm also helps solve a version of the robust stable matching problem. We discuss another potential application, namely obtaining new insights into the incentive compatibility properties of the Gale-Shapley Deferred Acceptance Algorithm.Comment: arXiv admin note: substantial text overlap with arXiv:1804.0553