  In this paper we study the approximability of (Finite-)Valued Constraint
Satisfaction Problems (VCSPs) with a fixed finite constraint language {\Gamma}
consisting of finitary functions on a fixed finite domain. An instance of VCSP
is given by a finite set of variables and a sum of functions belonging to
{\Gamma} and depending on a subset of the variables. Each function takes values
in [0, 1] specifying costs of assignments of labels to its variables, and the
goal is to find an assignment of labels to the variables that minimizes the
sum. A recent result of Ene et al. says that, under the mild technical
condition that {\Gamma} contains the equality relation, the basic LP relaxation
is optimal for constant-factor approximation for VCSP({\Gamma}) unless the
Unique Games Conjecture fails. Using the algebraic approach to the CSP, we give
new natural algebraic conditions for the finiteness of the integrality gap for
the basic LP relaxation of VCSP({\Gamma}). We also show how these algebraic
conditions can in principle be used to round solutions of the basic LP
relaxation, and how, for several examples that cover all previously known
cases, this leads to efficient constant-factor approximation algorithms.
Finally, we show that the absence of another algebraic condition leads to
NP-hardness of constant-factor approximation.
