822 research outputs found
The algebraic dichotomy conjecture for infinite domain Constraint Satisfaction Problems
We prove that an -categorical core structure primitively positively
interprets all finite structures with parameters if and only if some stabilizer
of its polymorphism clone has a homomorphism to the clone of projections, and
that this happens if and only if its polymorphism clone does not contain
operations , , satisfying the identity .
This establishes an algebraic criterion equivalent to the conjectured
borderline between P and NP-complete CSPs over reducts of finitely bounded
homogenous structures, and accomplishes one of the steps of a proposed strategy
for reducing the infinite domain CSP dichotomy conjecture to the finite case.
Our theorem is also of independent mathematical interest, characterizing a
topological property of any -categorical core structure (the existence
of a continuous homomorphism of a stabilizer of its polymorphism clone to the
projections) in purely algebraic terms (the failure of an identity as above).Comment: 15 page
Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem
The Algebraic Dichotomy Conjecture states that the Constraint Satisfaction
Problem over a fixed template is solvable in polynomial time if the algebra of
polymorphisms associated to the template lies in a Taylor variety, and is
NP-complete otherwise. This paper provides two new characterizations of
finitely generated Taylor varieties. The first characterization is using
absorbing subalgebras and the second one cyclic terms. These new conditions
allow us to reprove the conjecture of Bang-Jensen and Hell (proved by the
authors) and the characterization of locally finite Taylor varieties using weak
near-unanimity terms (proved by McKenzie and Mar\'oti) in an elementary and
self-contained way
On the reduction of the CSP dichotomy conjecture to digraphs
It is well known that the constraint satisfaction problem over general
relational structures can be reduced in polynomial time to digraphs. We present
a simple variant of such a reduction and use it to show that the algebraic
dichotomy conjecture is equivalent to its restriction to digraphs and that the
polynomial reduction can be made in logspace. We also show that our reduction
preserves the bounded width property, i.e., solvability by local consistency
methods. We discuss further algorithmic properties that are preserved and
related open problems.Comment: 34 pages. Article is to appear in CP2013. This version includes two
appendices with proofs of claims omitted from the main articl
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