We study the notion of robustness in stable matching problems. We first define robustness by introducing (a,b)-supermatches. An $(a,b)$-supermatch is a stable matching in which if $a$ pairs break up it is possible to find another stable matching by changing the partners of those $a$ pairs and at most $b$ other pairs. In this context, we define the most robust stable matching as a $(1,b)$-supermatch where b is minimum. We show that checking whether a given stable matching is a $(1,b)$-supermatch can be done in polynomial time. Next, we use this procedure to design a constraint programming model, a local search approach, and a genetic algorithm to find the most robust stable matching. Our empirical evaluation on large instances show that local search outperforms the other approaches.

We propose a constraint programming approach to the popular matching problem. We show that one can use the Global Cardinality Constraint to encode the problem even in cases that involve ties in the ordinal preferences of the applicants.

Different formulations are proposed, distinguishing The notion of popular matching was introduced by (Gardenfors between one-sided matching (Garg et al. 2010) and twosided 1975), but this notion has its roots in the 18th century matching, e.g. the stable marriage (SM) problem (Gale and the notion of a Condorcet winner.