Manipulating the outcome of stable marriage and roommates problems

The stable marriage and stable roommates problems have been extensively studied due to their applicability in various real-world scenarios. However, it might happen that no stable solution exists, or stable solutions do not meet certain requirements. In such cases, one might be interested in modifying the instance so that the existence of a stable outcome with the desired properties is ensured. We focus on three different modifications.

1. In the stable roommates problem, we show that finding a smallest subset of agents whose removal results in an instance with a stable matching is NP-complete if the capacities are greater than one, or the deleted agents must belong to a fixed subset of vertices. We further show that analogous results hold for the stable marriage problem when one would like to achieve the existence of a stable and perfect matching through the deletion of vertices.

2. We investigate how to modify the preferences of the agents as little as possible so that a given matching becomes stable. The deviation of the new preferences from the original ones can be measured in various ways; here, we concentrate on the ${\ell }_1$-norm. We show that, assuming the Unique Games Conjecture, the problem cannot be approximated within a factor smaller than 2. By relying on bipartite-submodular functions, we give a polynomial-time algorithm for the bipartite case. We also show that a similar approach leads to a 2-approximation for general graphs.

3. Last, we consider problems where the preferences of agents are not fully prescribed, and the goal is to decide whether the preference lists can be completed so that a stable matching exists. We settle the complexity of several variants, including cases when some of the edges are required to be included or excluded from the solution.