  The constraint satisfaction problem (CSP) on a relational structure B is to
decide, given a set of constraints on variables where the relations come from
B, whether or not there is a assignment to the variables satisfying all of the
constraints; the surjective CSP is the variant where one decides the existence
of a surjective satisfying assignment onto the universe of B. We present an
algebraic condition on the polymorphism clone of B and prove that it is
sufficient for the hardness of the surjective CSP on a finite structure B, in
the sense that this problem admits a reduction from a certain fixed-structure
CSP. To our knowledge, this is the first result that allows one to use
algebraic information from a relational structure B to infer information on the
complexity hardness of surjective constraint satisfaction on B. A corollary of
our result is that, on any finite non-trivial structure having only essentially
unary polymorphisms, surjective constraint satisfaction is NP-complete.
