  We discuss the connection between computational social choice (comsoc) and
computational complexity. We stress the work so far on, and urge continued
focus on, two less-recognized aspects of this connection. Firstly, this is very
much a two-way street: Everyone knows complexity classification is used in
comsoc, but we also highlight benefits to complexity that have arisen from its
use in comsoc. Secondly, more subtle, less-known complexity tools often can be
very productively used in comsoc.
