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 $M_1$, 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 develop an algorithmic technique to "propagate" changes through a stable matching. This technique is at the core of our 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 propagation technique allows for a fixed-parameter tractable algorithm with respect to the number of forbidden pairs when both forced and forbidden pairs are present. Moreover, we establish strong intractability results when preferences contain ties.