  In some preference aggregation scenarios, voters' preferences are highly
structured: e.g., the set of candidates may have one-dimensional structure (so
that voters' preferences are single-peaked) or be described by a binary
decision tree (so that voters' preferences are group-separable). However,
sometimes a single axis or a decision tree is insufficient to capture the
voters' preferences; rather, there is a small number $k$ of axes or decision
trees such that each vote in the profile is consistent with one of these axes
(resp., trees). In this work, we study the complexity of deciding whether
voters' preferences can be explained in this manner. For $k=2$, we use the
technique developed by Yang~[2020] in the context of single-peaked preferences
to obtain a polynomial-time algorithm for several domains: value-restricted
preferences, group-separable preferences, and a natural subdomain of
group-separable preferences, namely, caterpillar group-separable preferences.
For $k\ge 3$, the problem is known to be hard for single-peaked preferences; we
show that this is also the case for value-restricted and group-separable
preferences. Our positive results for $k=2$ make use of forbidden minor
characterizations of the respective domains; in particular, we establish that
the domain of caterpillar group-separable preferences admits a forbidden minor
characterization.
