  To any fixed, finite relational structure, $\mathbb{D}$, there is an
associated decision problem, CSP$(\mathbb{D})$, which is a restricted version
of the constraint satisfaction problem. In [8], the so called "algebraic
approach" to the constraint satisfaction problem was established. The authors
showed that to any finite relational structure, there is a corresponding finite
algebra, and that the complexity of CSP$(\mathbb{D})$ depends only on this
algebra. Therefore, they associate a decision problem, CSP$({\bf D})$ to an
algebra, ${\bf D}$, and ignore the relational structure. Their "algebraic
dichotomy conjecture" suggests that a technical condition on ${\bf D}$ implies
CSP$({\bf D})$ has a polynomial time algorithm. A significant sub-problem is
the case when some reduct of ${\bf D}$ has a congruence, $\theta$ so that ${\bf
D}/\theta$ has operations implying the local consistency algorithm correctly
solves CSP$({\bf D}/\theta)$, and each $\theta$-equivalence class, $B$, has
operations implying the few subpowers algorithm correctly solves CSP$({\bf
B})$. We give an algorithm for the case when ${\bf D}$ has a binary term
operation which is a $2$-semilattice operation on some quotient, ${\bf
D}/\theta$ of ${\bf D}$, a projection on each $\theta$-class, and two other
technical conditions are satisfied. Using this, we confirm the conjecture in
the case that ${\bf D}$ is in the join of two varieties, one of which has an
edge term and the other is term equivalent to the variety of $2$-semilattices.
