  Following the success of the so-called algebraic approach to the study of
decision constraint satisfaction problems (CSPs), exact optimization of valued
CSPs, and most recently promise CSPs, we propose an algebraic framework for
valued promise CSPs.
  To every valued promise CSP we associate an algebraic object, its so-called
valued minion. Our main result shows that the existence of a homomorphism
between the associated valued minions implies a polynomial-time reduction
between the original CSPs. We also show that this general reduction theorem
includes important inapproximability results, for instance, the
inapproximability of almost solvable systems of linear equations beyond the
random assignment threshold.
