1,656 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
QCSP on partially reflexive forests
We study the (non-uniform) quantified constraint satisfaction problem QCSP(H)
as H ranges over partially reflexive forests. We obtain a complexity-theoretic
dichotomy: QCSP(H) is either in NL or is NP-hard. The separating condition is
related firstly to connectivity, and thereafter to accessibility from all
vertices of H to connected reflexive subgraphs. In the case of partially
reflexive paths, we give a refinement of our dichotomy: QCSP(H) is either in NL
or is Pspace-complete
Faster Existential FO Model Checking on Posets
We prove that the model checking problem for the existential fragment of
first-order (FO) logic on partially ordered sets is fixed-parameter tractable
(FPT) with respect to the formula and the width of a poset (the maximum size of
an antichain). While there is a long line of research into FO model checking on
graphs, the study of this problem on posets has been initiated just recently by
Bova, Ganian and Szeider (CSL-LICS 2014), who proved that the existential
fragment of FO has an FPT algorithm for a poset of fixed width. We improve upon
their result in two ways: (1) the runtime of our algorithm is
O(f(|{\phi}|,w).n^2) on n-element posets of width w, compared to O(g(|{\phi}|).
n^{h(w)}) of Bova et al., and (2) our proofs are simpler and easier to follow.
We complement this result by showing that, under a certain
complexity-theoretical assumption, the existential FO model checking problem
does not have a polynomial kernel.Comment: Paper as accepted to the LMCS journal. An extended abstract of an
earlier version of this paper has appeared at ISAAC'14. Main changes to the
previous version are improvements in the Multicoloured Clique part (Section
4
Quantified Constraints in Twenty Seventeen
I present a survey of recent advances in the algorithmic and computational complexity theory of non-Boolean Quantified Constraint Satisfaction Problems, incorporating some more modern research directions
On The Relational Width of First-Order Expansions of Finitely Bounded Homogeneous Binary Cores with Bounded Strict Width
The relational width of a finite structure, if bounded, is always (1,1) or
(2,3). In this paper we study the relational width of first-order expansions of
finitely bounded homogeneous binary cores where binary cores are structures
with equality and some anti-reflexive binary relations such that for any two
different elements a, b in the domain there is exactly one binary relation R
with (a, b) in R.
Our main result is that first-order expansions of liberal finitely bounded
homogeneous binary cores with bounded strict width have relational width (2,
MaxBound) where MaxBound is the size of the largest forbidden substructure, but
is not less than 3, and liberal stands for structures that do not forbid
certain finite structures of small size. This result is built on a new approach
and concerns a broad class of structures including reducts of homogeneous
digraphs for which the CSP complexity classification has not yet been obtained.Comment: A long version of an extended abstract that appeared in LICS 202
QCSP on semicomplete digraphs
We study the (non-uniform) quantified constraint satisfaction
problem QCSP(H) as H ranges over semicomplete digraphs. We
obtain a complexity-theoretic trichotomy: QCSP(H) is either in P, is NP-complete or is Pspace-complete. The largest part of our work is the algebraic classification of precisely which semicompletes enjoy only essentially unary polymorphisms, which is combinatorially interesting in its own right
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