The power of primitive positive definitions with polynomially many variables

Lagerkvist, Victor; Wahlström, Magnus

doi:10.1093/logcom/exw005

Cited by 9 publications

(15 citation statements)

References 29 publications

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“…Finally, the study of which sequences of relations , of arity , have polynomial-sized (in ) pp-definitions in a finite constraint language Γ, has been addressed in [19]. Of course, this question for our relations plays a central role in this paper.…”

Section: Related Workmentioning

confidence: 99%

QCSP monsters and the demise of the chen conjecture

Zhuk

Martin

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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We give a surprising classification for the computational complexity of the Quantified Constraint Satisfaction Problem over a constraint language Γ, QCSP(Γ), where Γ 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 Γ such that QCSP(Γ) is DP-complete (from Boolean Hierarchy), and on a 10-element domain there exists a constraint language giving the complexity class Θ 2. Meanwhile, we prove the Chen Conjecture for finite conservative languages Γ. If the polymorphism clone of such Γ has the polynomially generated powers (PGP) property then QCSP(Γ) is in NP. Otherwise, the polymorphism clone of Γ has the exponentially generated powers (EGP) property and QCSP(Γ) is PSpace-complete.

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Section: Related Workmentioning

confidence: 99%

QCSP monsters and the demise of the chen conjecture

Zhuk

Martin

2020

Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing

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“…As discussed in the introduction, this is in contrast to the existing parameterized dichotomy results for CSP for more permissive parameters k [9,30,31,32,36], but it is in line with previous observations on the complexity of Inv(P ) for finite P ; cf. [33,Lemma 35]. To prove this we will use a padding procedure that, starting from a relation R of arity r, defines a padded relation R of arity r = r O (1) such that R is preserved by all partial operations of sufficiently small arity.…”

Section: Lower Boundsmentioning

confidence: 99%

Sparsification of SAT and CSP Problems via Tractable Extensions

Lagerkvist

Wahlström

2020

ACM Trans. Comput. Theory

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Unlike polynomial kernelization in general, for which many non-trivial results and methods exist, only few non-trival algorithms are known for polynomial-time sparsification. Furthermore, excepting problems on restricted inputs (such as graph problems on planar graphs), most such results rely upon encoding the instance as a system of bounded-degree polynomial equations. In particular, for SAT problems with a fixed constraint language Γ, every previously known result is captured by this approach, and for several such problems this is known to be tight. In this work, we investigate the limits of this approach-in particular, does it really cover all cases of non-trivial polynomial-time sparsification? We generalize the method using tools from the algebraic approach to constraint satisfaction problems (CSP). Every constraint which can be modelled via a system of linear equations, over some finite field F, also admits a finite domain extension to a tractable CSP with a Maltsev polymorphism, and using known algorithms for Maltsev languages we can show that every problem of the latter type admits a "basis" of O(n) constraints, which implies a linear sparsification for the original problem. This generalization appears to be strict; other special cases include constraints modelled via group equations over some finite group G. For sparsifications of polynomial but super-linear size we consider two extensions of this. Most directly, we can capture systems of bounded-degree polynomial equations in a "lift-and-project" manner, by finding Maltsev extensions for constraints over c-tuples of variables, for a basis with O(n c) constraints. Additionally, we may use extensions with k-edge polymorphisms instead of requiring a Maltsev polymorphism. We also investigate characterizations of when such extensions exist. We give an infinite sequence of partial polymorphisms φ1, φ2,. .. which characterizes whether a language Γ has a Maltsev extension (of possibly infinite domain). In the complementary direction of proving lower bounds on kernelizability, we prove that for any language not preserved by φ1, the corresponding SAT problem does not admit a kernel of size O(n 2−ε) for any ε > 0 unless the polynomial hierarchy collapses.

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“…Finally, the study of which sequences of relations R i , of arity i, have polynomial-sized (in i) pp-definitions in a finite constraint language Γ, has been addressed in [18]. Of course, this question for our relations τ i plays a central role in this paper.…”

Section: Related Workmentioning

confidence: 99%

QCSP monsters and the demise of the Chen Conjecture

Zhuk¹,

Martin²

2019

Preprint

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We give a surprising classification for the computational complexity of Quantified Constraint Satisfaction Problems, QCSP(Γ), where Γ is a finite language over 3 elements which contains all constants. In particular, such problems are either in P, NPcomplete, co-NP-complete or Pspace-complete. Our classification refutes the hitherto widely-believed Chen Conjecture.Additionally, we show that already on 4-element domain there exists a constraint language Γ such that QCSP(Γ) is DP-complete (from Boolean Hierarchy), and on 10element domain there exists a constraint language giving the complexity class Θ P 2 . Meanwhile, we prove the Chen Conjecture for finite conservative languages Γ. If the polymorphism clone of Γ has the polynomially generated powers (PGP) property then QCSP(Γ) is in NP. Otherwise, the polymorphism clone of Γ has the exponentially generated powers (EGP) property and QCSP(Γ) is Pspace-complete.

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The power of primitive positive definitions with polynomially many variables

Cited by 9 publications

References 29 publications

QCSP monsters and the demise of the chen conjecture

QCSP monsters and the demise of the chen conjecture

Sparsification of SAT and CSP Problems via Tractable Extensions

QCSP monsters and the demise of the Chen Conjecture

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