  We consider the problem of aggregating votes cast by a society on a fixed set
of issues, where each member of the society may vote for one of several
positions on each issue, but the combination of votes on the various issues is
restricted to a set of feasible voting patterns. We require the aggregation to
be supportive, i.e. for every issue $j$ the corresponding component $f_j$ of
every aggregator on every issue should satisfy $f_j(x_1, ,\ldots, x_n) \in
\{x_1, ,\ldots, x_n\}$. We prove that, in such a set-up, non-dictatorial
aggregation of votes in a society of some size is possible if and only if
either non-dictatorial aggregation is possible in a society of only two members
or a ternary aggregator exists that either on every issue $j$ is a majority
operation, i.e. the corresponding component satisfies $f_j(x,x,y) = f_j(x,y,x)
= f_j(y,x,x) =x, \forall x,y$, or on every issue is a minority operation, i.e.
the corresponding component satisfies $f_j(x,x,y) = f_j(x,y,x) = f_j(y,x,x) =y,
\forall x,y.$ We then introduce a notion of uniformly non-dictatorial
aggregator, which is defined to be an aggregator that on every issue, and when
restricted to an arbitrary two-element subset of the votes for that issue,
differs from all projection functions. We first give a characterization of sets
of feasible voting patterns that admit a uniformly non-dictatorial aggregator.
Then making use of Bulatov's dichotomy theorem for conservative constraint
satisfaction problems, we connect social choice theory with combinatorial
complexity by proving that if a set of feasible voting patterns $X$ has a
uniformly non-dictatorial aggregator of some arity then the multi-sorted
conservative constraint satisfaction problem on $X$, in the sense introduced by
Bulatov and Jeavons, with each issue representing a sort, is tractable;
otherwise it is NP-complete.
