  The stable marriage and stable roommates problems have been extensively
studied due to their high 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. In stable roommates problems with
all capacities being one, we give a simpler proof to show that removing an
agent from each odd cycle of a stable partition is optimal. We further show
that the problem becomes NP-complete if the capacities are greater than one, or
the deleted agents must belong to a fixed subset of vertices.
  Motivated by inverse optimization problems, 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 does not admit a better
than $2$ approximation. 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.
  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
extended 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.
