  When computing stable matchings, it is usually assumed that the preferences
of the agents in the matching market are fixed. However, in many realistic
scenarios, preferences change over time. Consequently, an initially stable
matching may become unstable. Then, a natural goal is to find a matching which
is stable with respect to the modified preferences and as close as possible to
the initial one. For Stable Marriage/Roommates, this problem was formally
defined as Incremental Stable Marriage/Roommates by Bredereck et al. [AAAI
'20]. As they showed that Incremental Stable Roommates and Incremental Stable
Marriage with Ties are NP-hard, we focus on the parameterized complexity of
these problems. We answer two open questions of Bredereck et al. [AAAI '20]: We
show that Incremental Stable Roommates is W[1]-hard parameterized by the number
of changes in the preferences, yet admits an intricate XP-algorithm, and we
show that Incremental Stable Marriage with Ties is W[1]-hard parameterized by
the number of ties. Furthermore, we analyze the influence of the degree of
"similarity" between the agents' preference lists, identifying several
polynomial-time solvable and fixed-parameter tractable cases, but also proving
that Incremental Stable Roommates and Incremental Stable Marriage with Ties
parameterized by the number of different preference lists are W[1]-hard.
