  We study the computational complexity of exact minimisation of
rational-valued discrete functions. Let $\Gamma$ be a set of rational-valued
functions on a fixed finite domain; such a set is called a finite-valued
constraint language. The valued constraint satisfaction problem,
$\operatorname{VCSP}(\Gamma)$, is the problem of minimising a function given as
a sum of functions from $\Gamma$. We establish a dichotomy theorem with respect
to exact solvability for all finite-valued constraint languages defined on
domains of arbitrary finite size.
  We show that every constraint language $\Gamma$ either admits a binary
symmetric fractional polymorphism in which case the basic linear programming
relaxation solves any instance of $\operatorname{VCSP}(\Gamma)$ exactly, or
$\Gamma$ satisfies a simple hardness condition that allows for a
polynomial-time reduction from Max-Cut to $\operatorname{VCSP}(\Gamma)$.
