8,753 research outputs found
The complexity of general-valued CSPs seen from the other side
The constraint satisfaction problem (CSP) is concerned with homomorphisms
between two structures. For CSPs with restricted left-hand side structures, the
results of Dalmau, Kolaitis, and Vardi [CP'02], Grohe [FOCS'03/JACM'07], and
Atserias, Bulatov, and Dalmau [ICALP'07] establish the precise borderline of
polynomial-time solvability (subject to complexity-theoretic assumptions) and
of solvability by bounded-consistency algorithms (unconditionally) as bounded
treewidth modulo homomorphic equivalence.
The general-valued constraint satisfaction problem (VCSP) is a generalisation
of the CSP concerned with homomorphisms between two valued structures. For
VCSPs with restricted left-hand side valued structures, we establish the
precise borderline of polynomial-time solvability (subject to
complexity-theoretic assumptions) and of solvability by the -th level of the
Sherali-Adams LP hierarchy (unconditionally). We also obtain results on related
problems concerned with finding a solution and recognising the tractable cases;
the latter has an application in database theory.Comment: v2: Full version of a FOCS'18 paper; improved presentation and small
correction
Canonical Polymorphisms of Ramsey Structures and the Unique Interpolation Property
Constraint satisfaction problems for first-order reducts of finitely bounded
homogeneous structures form a large class of computational problems that might
exhibit a complexity dichotomy, P versus NP-complete. A powerful method to
obtain polynomial-time tractability results for such CSPs is a certain
reduction to polynomial-time tractable finite-domain CSPs defined over k-types,
for a sufficiently large k. We give sufficient conditions when this method can
be applied and illustrate how to use the general results to prove a new
complexity dichotomy for first-order expansions of the basic relations of the
spatial reasoning formalism RCC5
Finite Algebras with Hom-Sets of Polynomial Size
We provide an internal characterization of those finite algebras (i.e.,
algebraic structures) such that the number of homomorphisms from
any finite algebra to is bounded from above by a
polynomial in the size of . Namely, an algebra has this
property if, and only if, no subalgebra of has a nontrivial
strongly abelian congruence. We also show that the property can be decided in
polynomial time for algebras in finite signatures. Moreover, if is
such an algebra, the set of all homomorphisms from to
can be computed in polynomial time given as input. As an
application of our results to the field of computational complexity, we
characterize inherently tractable constraint satisfaction problems over fixed
finite structures, i.e., those that are tractable and remain tractable after
expanding the fixed structure by arbitrary relations or functions
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
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