  The complexity and approximability of the constraint satisfaction problem
(CSP) has been actively studied over the last 20 years. A new version of the
CSP, the promise CSP (PCSP) has recently been proposed, motivated by open
questions about the approximability of variants of satisfiability and graph
colouring. The PCSP significantly extends the standard decision CSP. The
complexity of CSPs with a fixed constraint language on a finite domain has
recently been fully classified, greatly guided by the algebraic approach, which
uses polymorphisms --- high-dimensional symmetries of solution spaces --- to
analyse the complexity of problems. The corresponding classification for PCSPs
is wide open and includes some long-standing open questions, such as the
complexity of approximate graph colouring, as special cases.
  The basic algebraic approach to PCSP was initiated by Brakensiek and
Guruswami, and in this paper we significantly extend it and lift it from
concrete properties of polymorphisms to their abstract properties. We introduce
a new class of problems that can be viewed as algebraic versions of the (Gap)
Label Cover problem, and show that every PCSP with a fixed constraint language
is equivalent to a problem of this form. This allows us to identify a "measure
of symmetry" that is well suited for comparing and relating the complexity of
different PCSPs via the algebraic approach. We demonstrate how our theory can
be applied by improving the state-of-the-art in approximate graph colouring: we
show that, for any $k\geq 3$, it is NP-hard to find a $(2k-1)$-colouring of a
given $k$-colourable graph.
