A Simple Algorithm for Mal'tsev Constraints

Generalized Majority-Minority Operations are Tractable

“…In [2] (see [7] for a simpler proof), it was shown that for every Malt'sev operation ϕ, CSP(Inv(ϕ)) is solvable in polynomial time. This general result encompasses some previously known tractable cases of the CSP, such as CSP with constraints defined by a system of linear equations [23] or CSP with near-subgroups and its cosets [19,18].…”

SDP Gaps for 2-to-1 and Other Label-Cover Variants

“…Deciding whether a system of linear equations is completely satisfiable is of course in P. Alternatively, one can note that constraints x − y ∈ {α, α + γ} mod F k 2 are Mal'tsev constraints, and hence deciding satisfiability of CSPs based on them is in P by the work of Bulatov and Dalmau [1].…”

Recent Results on the Algebraic Approach to the CSP

“…Generalizing this, Bulatov [5] proves that if a finite algebra A has a term p(x, y, z) that satisfies the equations p(x, x, y) ≈ p(y, x, x) ≈= y for all x, y ∈ A then A is also tractable (any operation that satisfies these equations is known as a Mal'tsev operation, see Example 4). The proof of this theorem found in [13] exploits the fact that any finite algebra with a Mal'tsev term has the small generating sets property (and hence, few subpowers).…”

“…In a modification of the algorithm presented in [13], Dalmau shows in [22] that any finite algebra that has a GMM term is tractable. As in the case of algebras with Mal'tsev terms, these algebras have few subpowers and the small generating sets property and it is this latter property that plays a crucial role in the proof.…”

“…Example 4 ( [5,13,20,22,29,31] • a semilattice operation, that is a binary operation f satisfying the equations:…”