Adapting Stable Matchings to Forced and Forbidden Pairs
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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.
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