= \bigcup_{i \in [n], |h_i(a)| = 1} h_i(a)$. Since $g$ is a homomorphism from ${\mathbf{A}}$ to ${\mathsf{UnionSing}}({\wp}({\mathbf{B}})^{n})$, we can compose $g$ and a homomorphism from ${\mathsf{UnionSing}}({\wp}({\mathbf{B}})^{n})$ to ${\mathbf{B}}$, which we know to exist by assumption, to get a homomorphism from ${\mathbf{A}}$ to ${\mathbf{B}}$. Consequently, singleton arc consistency solves ${{\mathsf{CSP}}}({\mathbf{B}})$.

Strength Comparison {#sect:comparison}
===================

In this section, we investigate relationships among the sets of structures solvable by the various algorithms presented. We show that for the structures having all constants, AC solves a strictly smaller set of structures than LAAC does; on the other hand, we show that there is a structure (not having all constants) solvable by AC but not LAAC. We then show that the structures solvable by AC or LAAC are strictly contained in those solvable by PAC; and, in turn, that the structures solvable by PAC are strictly contained in those solvable by SAC. We also show that the structures solvable by SAC (and hence, those solvable by any of the studied algorithms) all fall into the class of structures having *bounded width*; bounded width is a well-studied condition admitting multiple characterizations [@FederVardi; @LaroseZadori; @bkbw].

\[prop:all-constants-ac-implies-laac\] Suppose that ${\mathbf{B}}$ is a structure having all constants. If ${{\mathsf{CSP}}}({\mathbf{B}})$ 