Adapting Stable Matchings to Forced and Forbidden Pairs

TL;DR

This paper addresses the problem of modifying stable matchings to incorporate forced pairs and exclude forbidden pairs, providing polynomial-time algorithms for some cases and proving NP-hardness for others, with extensions to tie preferences.

Contribution

It introduces a new problem of adapting stable matchings with forced and forbidden pairs, offering polynomial algorithms for certain cases and complexity results for others.

Findings

Polynomial-time algorithm for adapting Stable Roommates to forced pairs.

NP-hardness of adapting to forbidden pairs.

Fixed-parameter tractability when both forced and forbidden pairs are present.

Abstract

We introduce the problem of adapting a stable matching to forced and forbidden pairs. Specifically, given a stable matching $M_1$, a set $Q$ of forced pairs, and a set $P$ of forbidden pairs, we want to find a stable matching that includes all pairs from $Q$, no pair from $P$, and that is as close as possible to $M_1$. We study this problem in four classical stable matching settings: Stable Roommates (with Ties) and Stable Marriage (with Ties). As our main contribution, we employ the theory of rotations for Stable Roommates to develop a polynomial-time algorithm for adapting Stable Roommates matchings to forced pairs. In contrast to this, we show that the same problem for forbidden pairs is NP-hard. However, our polynomial-time algorithm for the case of only forced pairs can be extended to a fixed-parameter tractable algorithm with respect to the number of forbidden pairs when both forced and forbidden pairs are present. Moreover, we also study the setting where preferences contain ties. Here, depending on the chosen stability criterion, we show either that our algorithmic results can be extended or that formerly tractable problems become intractable.