On the Complexity of Robust Stable Marriage

Robust Stable Marriage (RSM) is a variant of the classical Stable Marriage problem, where the robustness of a given stable matching is measured by the number of modifications required for repairing it in case an unforeseen event occurs. We focus on the complexity of finding an (a, b)-supermatch. An (a, b)-supermatch is defined as a stable matching in which if any a (non-fixed) men/women break up it is possible to find another stable matching by changing the partners of those a men/women and also the partners of at most b other couples. In order to show deciding if there exists an (a, b)-supermatch is $\mathcal{NP}$-complete, we first introduce a SAT formulation that is $\mathcal{NP}$-complete by using Schaefer’s Dichotomy Theorem. Then, we show the equivalence between the SAT formulation and finding a (1, 1)-supermatch on a specific family of instances.