The complexity of general-valued CSPs seen from the other side

General-valued constraint satisfaction problems (VCSPs) are generalisations of CSPs dealing with homomorphisms between two valued structures. We study the complexity of structural restrictions for VCSPs, that is, restrictions defined by classes of admissible lefthand side valued structures. As our main result, we show that for VCSPs of bounded arity the only tractable structural restrictions are those of bounded treewidth modulo valued equivalence, thus identifying the precise borderline of tractability. Our result generalises a result of Dalmau, Kolaitis, and Vardi [CP’02] and Grohe [JACM’07] showing that for CSPs of bounded arity the tractable restrictions are precisely those with bounded treewidth modulo homomorphic equivalence. All the tractable restrictions we identify are solved by the well-known Sherali-Adams LP hierarchy. We take a closer look into this hierarchy and study the power of Sherali-Adams for solving VCSPs. Our second result is a precise characterisation of the left-hand side valued structures solved to optimality by the k-th level of the Sherali-Adams LP hierarchy.