Theory of and Experiments on Minimally Invasive Stability Preservation in Changing Two-Sided Matching Markets

Abstract

Following up on purely theoretical work, we contribute further theoretical insights into adapting stable two-sided matchings to change. Moreover, we perform extensive empirical studies hinting at numerous practically useful properties. Our theoretical extensions include the study of new problems (that is, incremental variants of Almost Stable Marriage and Hospital Residents), focusing on their (parameterized) computational complexity and the equivalence of various change types (thus simplifying algorithmic and complexity-theoretic studies for various natural change scenarios). Our experimental findings reveal, for instance, that allowing the new matching to be blocked by a few pairs significantly decreases the difference between the old and the new matching.

Deepening the (Parameterized) Complexity Analysis of Incremental Stable Matching Problems

“…For instance, we consider the case where all but s agents have the same preference list (we call these s agents outliers), or the case where each agent has one out of only p different master preference lists. We devise an algorithm that enumerates all stable matchings in an SR instance in FPT time with respect to s, implying an FPT algorithm for ISR parameterized by s. In contrast to this and to a simple FPT algorithm for the number of agent types [5], we prove that ISR and ISM-T are W [1]-hard with respect to the number p of different preference lists.…”

“…1 Notably, by the equivalence theorem of Boehmer et al [5,Theorem 1], all our results (except for Theorem 3 where the constant |P1 ⊕ P2| increases by a small number) still hold if |P1 ⊕ P2| instead denotes the number of agents whose preferences changed, the number of deleted agents (i.e., the number of agents with empty preferences in P2 and non-empty preferences in P1), or the number of added agents (i.e., the number of agents with empty preferences in P1 and non-empty preferences in P2).…”

“…Two agents are of the same type if they have the same preferences and all other agents are indifferent between them. Notably, Boehmer et al [5,Proposition 5] proved that ISM-T is fixed-parameter tractable with respect to the number of agent types. Their algorithm also works for ISR-T.…”

“…The closeness condition here is due to the fact that in most applications reassignments come at some cost which we want to minimize (e.g., in the above New York example, reassigning students might make it necessary for the family to reallocate within the city). We build upon the work of Bredereck et al [7], who performed a first systematic study of incremental versions of stable matching problems, and the recent (partially empirical) follow-up work by Boehmer et al [5], who proved among others that different types of changes are "equivalent" to each other. The central focus of our studies lies on the Stable Roommates (SR) problem: given a set of agents with each agent having preferences over other agents, the task is to find a stable matching, i.e., a matching so that there are no two agents preferring each other to their assigned partner.…”

“…As ISR-T can be considered as a generalization of ISM-T, their results motivate us to focus on the NP-hard ISR and ISM-T problems (which are somewhat incomparable problems). Recently, Boehmer et al [5] followed up on the work of Bredereck et al [7], proving that different types of changes such as deleting agents or performing swaps of adjacent agents in some preference list are "equivalent". Moreover, they introduced incremental variants of further stable matching problems and performed empirical studies.…”