Jointly Stable Matchings

In the stable marriage problem, we are given a set of men, a set of women, and each person\u27s preference list. Our task is to find a stable matching, that is, a matching admitting no unmatched (man, woman)-pair each of which improves the situation by being matched together. It is known that any instance admits at least one stable matching. In this paper, we consider a natural extension where k (>= 2) sets of preference lists L_i (1 <= i <= k) over the same set of people are given, and the aim is to find a jointly stable matching, a matching that is stable with respect to all L_i. We show that the decision problem is NP-complete already for k=2, even if each person\u27s preference list is of length at most four, while it is solvable in linear time for any k if each man\u27s preference list is of length at most two (women\u27s lists can be of unbounded length). We also show that if each woman\u27s preference lists are same in all L_i, then the problem can be solved in linear time

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

Popular and Dominant Matchings with Uncertain, Multilayer and Aggregated Preferences

We study the Popular Matching problem in multiple models, where the preferences of the agents in the instance may change or may be unknown/uncertain. In particular, we study an Uncertainty model, where each agent has a possible set of preferences, a Multilayer model, where there are layers of preference profiles, a Robust model, where any agent may move some other agents up or down some places in his preference list and an Aggregated Preference model, where votes are summed over multiple instances with different preferences. We study both one-sided and two-sided preferences in bipartite graphs. In the one-sided model, we show that all our problems can be solved in polynomial time by utilizing the structure of popular matchings. We also obtain nice structural results. With two-sided preferences, we show that all four above models lead to NP-hard questions for popular matchings. By utilizing the connection between dominant matchings and stable matchings, we show that in the robust and uncertainty model, a certainly dominant matching in all possible prefernce profiles can be found in polynomial-time, whereas in the multilayer and aggregated models, the problem remains NP-hard for dominant matchings too. We also answer an open question about $d$-robust stable matchings

Adapting Stable Matchings to Evolving Preferences

Adaptivity to changing environments and constraints is key to success in modern society. We address this by proposing "incrementalized versions" of Stable Marriage and Stable Roommates. That is, we try to answer the following question: for both problems, what is the computational cost of adapting an existing stable matching after some of the preferences of the agents have changed. While doing so, we also model the constraint that the new stable matching shall be not too different from the old one. After formalizing these incremental versions, we provide a fairly comprehensive picture of the computational complexity landscape of Incremental Stable Marriage and Incremental Stable Roommates. To this end, we exploit the parameters "degree of change" both in the input (difference between old and new preference profile) and in the output (difference between old and new stable matching). We obtain both hardness and tractability results, in particular showing a fixed-parameter tractability result with respect to the parameter "distance between old and new stable matching".Comment: Accepted to AAAI 202

Jointly stable matchings

In the stable marriage problem, we are given a set of men, a set of women, and each person’s preference list. Our task is to find a stable matching, that is, a matching admitting no unmatched (man, woman)-pair each of which improves the situation by being matched together. It is known that any instance admits at least one stable matching. In this paper, we consider a natural extension where k( ≥ 2) sets of preference lists Li (1≤ i ≤ k ) over the same set of people are given, and the aim is to find a jointly stable matching, a matching that is stable with respect to all Li . We show that the decision problem is NP-complete for the following two restricted cases; (1) k = 2 and each person’s preference list is of length at most four, and (2) k = 4 , each man’s preference list is of length at most three, and each woman’s preference list is of length at most four. On the other hand, we show that it is solvable in linear time for any k if each man’s preference list is of length at most two (women’s lists can be of unbounded length). We also show that if each woman’s preference lists are same in all Li , then the problem can be solved in linear time