Stable Marriage with Groups of Similar Agents

Abstract

Many important stable matching problems are known to be NP-hard, even when strong restrictions are placed on the input. In this paper we seek to identify structural properties of instances of stable matching problems which will allow us to design efficient algorithms using elementary techniques. We focus on the setting in which all agents involved in some matching problem can be partitioned into k different types, where the type of an agent determines his or her preferences, and agents have preferences over types (which may be refined by more detailed preferences within a single type). This situation would arise in practice if agents form preferences solely based on some small collection of agents' attributes. We also consider a generalisation in which each agent may consider some small collection of other agents to be exceptional, and rank these in a way that is not consistent with their types; this could happen in practice if agents have prior contact with a small number of candidates. We show that (for the case without exceptions), the well-known NP-hard matching problem Max SMTI (that of finding the maximum cardinality stable matching in an instance of stable marriage with ties and incomplete lists) belongs to the parameterised complexity class FPT when parameterised by the number of different types of agents needed to describe the instance. This tractability result can be extended to the setting in which each agent promotes at most one "exceptional" candidate to the top of his/her list (when preferences within types are not refined), but the problem remains NP-hard if preference lists can contain two or more exceptions and the exceptional candidates can be placed anywhere in the preference lists.

Stable roommates with narcissistic, single-peaked, and single-crossing preferences

“…First, to better understand the NPcompleteness result, one may study the parameterized complexity with respect to the "degree" of incompleteness of the input preferences, such as the number of ties or the number of agents that are in the same equivalence class of the tie-relation. We refer to some recent papers on the parameterized complexity of preference-based stable matching problems [1,11,12,[14][15][16]29,[40][41][42]45] for this line of research. Second, we were not able to settle the computational complexity for complete preferences that are also single-peaked and single-crossing.…”

Parameterized complexity of stable roommates with ties and incomplete lists through the lens of graph parameters

“…They observed that it is NP-hard even if the maximum length of a tie is constant but showed that Max-SRTI is fixed-parameter tractable when parameterized by the combined parameter "number of ties and maximum length of a tie". Meeks and Rastegari [30] considered a setting where the agents are partitioned into different types having the same preferences. They show that the problem is FPT in the number of types.…”