Commutative, idempotent groupoids and the constraint satisfaction problem

A restatement of the Algebraic Dichotomy Conjecture, due to Marti and McKenzie, postulates that if a finite algebra possesses a weak near-unanimity term, then the corresponding constraint satisfaction problem is tractable. A binary operation is weak near-unanimity if and only if it is both commutative and idempotent. Thus if the dichotomy conjecture is true, any finite commutative, idempotent groupoid (CI-groupoid) will be tractable. It is known that every semilattice (associative CI-groupoid) is tractable. A groupoid identity is of Bol-Moufang type if the same three variables appear on either side, one of the variables is repeated, the remaining two variables appear once, and the variables appear in the same order on either side (for example, x(x(yz))=(x(xy))z). These identities can be thought of as generalizations of associativity. We show that there are exactly 8 varieties of CI-groupoids defined by a single additional identity of Bol-Moufang type, derive some of their important structural properties, and use that structure theory to show that 7 of the varieties are tractable. We also characterize the finite members of the variety of CI-groupoids satisfying the self-distributive law x(yz)=(xy)(xz), and show that they are tractable. Varieties of CI-groupoids satisfying other identities strictly weaker than associativity are also considered, and shown to be tractable

On Backdoors to Tractable Constraint Languages

International audienceIn the context of CSPs, a strong backdoor is a subset of variables such that every complete assignment yields a residual instance guaranteed to have a specified property. If the property allows efficient solving, then a small strong backdoor provides a reasonable decomposition of the original instance into easy instances. An important challenge is the design of algorithms that can find quickly a small strong backdoor if one exists. We present a systematic study of the parameterized complexity of backdoor detection when the target property is a restricted type of constraint language defined by means of a family of polymor-phisms. In particular, we show that under the weak assumption that the polymorphisms are idempotent, the problem is unlikely to be FPT when the parameter is either r (the constraint arity) or k (the size of the backdoor) unless P = NP or FPT = W[2]. When the parameter is k + r, however, we are able to identify large classes of languages for which the problem of finding a small backdoor is FPT

Connectedness in Cayley Graphs and P/NP Dichotomy for Quay Algebras

This senior thesis attempts to determine the extent to which the P/NP dichotomy of finite algebras (as proven by Bulatov, et.al in 2017) can be cast in terms of connectedness in Cayley graphs. This research is motivated by Prof. Robert McGrail\u27s work ``CSPs and Connectedness: P/NP-Complete Dichotomy for Idempotent, Right Quasigroups published in 2014 in which he demonstrates the strong correspondence between tractability and total path-connectivity in Cayley graphs for right, idempotent quasigroups. In particular, we will introduce the notion of total V-connectedness and show how it could be potentially used to phrase the dichotomy in terms of connectivity for another class of algebras, namely for Quay algebras

Generalized Majority-Minority Operations are Tractable

Generalized majority-minority (GMM) operations are introduced as a common generalization of near unanimity operations and Mal'tsev operations on finite sets. We show that every instance of the constraint satisfaction problem (CSP), where all constraint relations are invariant under a (fixed) GMM operation, is solvable in polynomial time. This constitutes one of the largest tractable cases of the CSP