89 research outputs found
Conservative constraint satisfaction re-revisited
Conservative constraint satisfaction problems (CSPs) constitute an important
particular case of the general CSP, in which the allowed values of each
variable can be restricted in an arbitrary way. Problems of this type are well
studied for graph homomorphisms. A dichotomy theorem characterizing
conservative CSPs solvable in polynomial time and proving that the remaining
ones are NP-complete was proved by Bulatov in 2003. Its proof, however, is
quite long and technical. A shorter proof of this result based on the absorbing
subuniverses technique was suggested by Barto in 2011. In this paper we give a
short elementary prove of the dichotomy theorem for the conservative CSP
Galois correspondence for counting quantifiers
We introduce a new type of closure operator on the set of relations,
max-implementation, and its weaker analog max-quantification. Then we show that
approximation preserving reductions between counting constraint satisfaction
problems (#CSPs) are preserved by these two types of closure operators.
Together with some previous results this means that the approximation
complexity of counting CSPs is determined by partial clones of relations that
additionally closed under these new types of closure operators. Galois
correspondence of various kind have proved to be quite helpful in the study of
the complexity of the CSP. While we were unable to identify a Galois
correspondence for partial clones closed under max-implementation and
max-quantification, we obtain such results for slightly different type of
closure operators, k-existential quantification. This type of quantifiers are
known as counting quantifiers in model theory, and often used to enhance first
order logic languages. We characterize partial clones of relations closed under
k-existential quantification as sets of relations invariant under a set of
partial functions that satisfy the condition of k-subset surjectivity. Finally,
we give a description of Boolean max-co-clones, that is, sets of relations on
{0,1} closed under max-implementations.Comment: 28 pages, 2 figure
The Subpower Membership Problem for Finite Algebras with Cube Terms
The subalgebra membership problem is the problem of deciding if a given
element belongs to an algebra given by a set of generators. This is one of the
best established computational problems in algebra. We consider a variant of
this problem, which is motivated by recent progress in the Constraint
Satisfaction Problem, and is often referred to as the Subpower Membership
Problem (SMP). In the SMP we are given a set of tuples in a direct product of
algebras from a fixed finite set of finite algebras, and are
asked whether or not a given tuple belongs to the subalgebra of the direct
product generated by a given set.
Our main result is that the subpower membership problem SMP() is
in P if is a finite set of finite algebras with a cube term,
provided is contained in a residually small variety. We also
prove that for any finite set of finite algebras in a variety
with a cube term, each one of the problems SMP(), SMP(), and finding compact representations for subpowers in
, is polynomial time reducible to any of the others, and the first
two lie in NP
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