187 research outputs found
QCSP monsters and the demise of the chen conjecture.
We give a surprising classification for the computational complexity
of the Quantified Constraint Satisfaction Problem over a constraint
language Î, QCSP(Î), where Î is a finite language over 3 elements
which contains all constants. In particular, such problems are either in P, NP-complete, co-NP-complete or PSpace-complete. Our
classification refutes the hitherto widely-believed Chen Conjecture.
Additionally, we show that already on a 4-element domain there
exists a constraint language Î such that QCSP(Î) is DP-complete
(from Boolean Hierarchy), and on a 10-element domain there exists
a constraint language giving the complexity class Î
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2
.
Meanwhile, we prove the Chen Conjecture for finite conservative languages Î. If the polymorphism clone of such Î has the
polynomially generated powers (PGP) property then QCSP(Î) is in
NP. Otherwise, the polymorphism clone of Î has the exponentially
generated powers (EGP) property and QCSP(Î) is PSpace-complete
The complexity of quantified constraints using the algebraic formulation
Peer reviewedFinal Published versio
From Complexity to Algebra and Back: Digraph Classes, Collapsibility, and the PGP
Inspired by computational complexity results for the quantified constraint satisfaction problem, we study the clones of idem potent polymorphisms of certain digraph classes. Our first results are two algebraic dichotomy, even "gap", theorems. Building on and extending [Martin CP'11], we prove that partially reflexive paths bequeath a set of idem potent polymorphisms whose associated clone algebra has: either the polynomially generated powers property (PGP), or the exponentially generated powers property (EGP). Similarly, we build on [DaMM ICALP'14] to prove that semi complete digraphs have the same property. These gap theorems are further motivated by new evidence that PGP could be the algebraic explanation that a QCSP is in NP even for unbounded alternation. Along the way we also effect a study of a concrete form of PGP known as collapsibility, tying together the algebraic and structural threads from [Chen Sicomp'08], and show that collapsibility is equivalent to its Pi2-restriction. We also give a decision procedure for k-collapsibility from a singleton source of a finite structure (a form of collapsibility which covers all known examples of PGP for finite structures). Finally, we present a new QCSP trichotomy result, for partially reflexive paths with constants. Without constants it is known these QCSPs are either in NL or Pspace-complete [Martin CP'11], but we prove that with constants they attain the three complexities NL, NP-complete and Pspace-complete
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
An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction
We prove an algebraic preservation theorem for positive Horn definability in
aleph-zero categorical structures. In particular, we define and study a
construction which we call the periodic power of a structure, and define a
periomorphism of a structure to be a homomorphism from the periodic power of
the structure to the structure itself. Our preservation theorem states that,
over an aleph-zero categorical structure, a relation is positive Horn definable
if and only if it is preserved by all periomorphisms of the structure. We give
applications of this theorem, including a new proof of the known complexity
classification of quantified constraint satisfaction on equality templates
Generic Expression Hardness Results for Primitive Positive Formula Comparison
We study the expression complexity of two basic problems involving the
comparison of primitive positive formulas: equivalence and containment. In
particular, we study the complexity of these problems relative to finite
relational structures. We present two generic hardness results for the studied
problems, and discuss evidence that they are optimal and yield, for each of the
problems, a complexity trichotomy
Growth of generating sets for direct powers of classical algebraic structures
For an algebraic structure A denote by d(A) the smallest size of a generating set for A, and let d(A)=(d(A),d(A2),d(A3),âŠ), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d(A) when A is one of the classical structuresâa group, ring, module, algebra or Lie algebra. We show that if A is finite then d(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.Publisher PDFPeer reviewe
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