818 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
Hybrid VCSPs with crisp and conservative valued templates
A constraint satisfaction problem (CSP) is a problem of computing a
homomorphism between two relational
structures. Analyzing its complexity has been a very fruitful research
direction, especially for fixed template CSPs, denoted , in
which the right side structure is fixed and the left side
structure is unconstrained.
Recently, the hybrid setting, written ,
where both sides are restricted simultaneously, attracted some attention. It
assumes that is taken from a class of relational structures
that additionally is closed under inverse homomorphisms. The last
property allows to exploit algebraic tools that have been developed for fixed
template CSPs. The key concept that connects hybrid CSPs with fixed-template
CSPs is the so called "lifted language". Namely, this is a constraint language
that can be constructed from an input . The
tractability of that language for any input is a
necessary condition for the tractability of the hybrid problem.
In the first part we investigate templates for which the
latter condition is not only necessary, but also is sufficient. We call such
templates widely tractable. For this purpose, we construct from
a new finite relational structure and define
as a class of structures homomorphic to . We
prove that wide tractability is equivalent to the tractability of
. Our proof is based on the key observation
that is homomorphic to if and only if the core of
is preserved by a Siggers polymorphism. Analogous
result is shown for valued conservative CSPs.Comment: 21 pages. arXiv admin note: text overlap with arXiv:1504.0706
The complexity of the list homomorphism problem for graphs
We completely classify the computational complexity of the list H-colouring
problem for graphs (with possible loops) in combinatorial and algebraic terms:
for every graph H the problem is either NP-complete, NL-complete, L-complete or
is first-order definable; descriptive complexity equivalents are given as well
via Datalog and its fragments. Our algebraic characterisations match important
conjectures in the study of constraint satisfaction problems.Comment: 12 pages, STACS 201
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