Adapting Stable Matchings to Forced and Forbidden Pairs

We introduce the problem of adapting a stable matching to forced and forbidden pairs. Specifically, given a stable matching M₁, 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 M₁. 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.