Online 2-stage Stable Matching

We focus on an online 2-stage problem, motivated by the following situation: consider a system where students shall be assigned to universities. There is a first round where some students apply, and a first (stable) matching $M_1$ has to be computed. However, some students may decide to leave the system (change their plan, go to a foreign university, or to some institution not in the system). Then, in a second round (after these deletions), we shall compute a second (final) stable matching $M_2$. As it is undesirable to change assignments, the goal is to minimize the number of divorces/modifications between the two stable matchings $M_1$ and $M_2$. Then, how should we choose $M_1$ and $M_2$? We show that there is an {\it optimal online} algorithm to solve this problem. In particular, thanks to a dominance property, we show that we can optimally compute $M_1$ without knowing the students that will leave the system. We generalize the result to some other possible modifications in the input (students, open positions). We also tackle the case of more stages, showing that no competitive (online) algorithm can be achieved for the considered problem as soon as there are 3 stages.