A Constraint Satisfaction Problem Algorithm for Certain 2-Semilattice-over-Edge Algebras

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.