  An elegant characterization of the complexity of constraint satisfaction
problems has emerged in the form of the the algebraic dichotomy conjecture of
[BKJ00]. Roughly speaking, the characterization asserts that a CSP {\Lambda} is
tractable if and only if there exist certain non-trivial operations known as
polymorphisms to combine solutions to {\Lambda} to create new ones. In an
entirely separate line of work, the unique games conjecture yields a
characterization of approximability of Max-CSPs. Surprisingly, this
characterization for Max-CSPs can also be reformulated in the language of
polymorphisms.
  In this work, we study whether existence of non-trivial polymorphisms implies
tractability beyond the realm of constraint satisfaction problems, namely in
the value-oracle model. Specifically, given a function f in the value-oracle
model along with an appropriate operation that never increases the value of f ,
we design algorithms to minimize f . In particular, we design a randomized
algorithm to minimize a function f that admits a fractional polymorphism which
is measure preserving and has a transitive symmetry.
  We also reinterpret known results on MaxCSPs and thereby reformulate the
unique games conjecture as a characterization of approximability of max-CSPs in
terms of their approximate polymorphisms.
