The Complexity of General-Valued CSPs

Kolmogorov, Vladimir; Krokhin, Andrei; Rolínek, Michal

doi:10.1137/16m1091836

Cited by 54 publications

(62 citation statements)

References 40 publications

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“…In one of the books of reference in the field [4], model theory is described as algebra+logic. Working in this same framework, and in the line of recent works taking an algebraic approach to valued CSP (see for instance [16] and [17]), we have presented an algebraic characterization of the preservation of ppformulas in terms of direct products and homomorphisms. Theorem 2 tells us that there is a good trade-off between algebra and logic in the fuzzy positiveprimitive fragment.…”

Section: Discussion and Future Workmentioning

confidence: 99%

“…However, the notion of fuzzy homomorphism traditionally used in the fuzzy literature, do not encompass other notions of polymorphism such as weighted or fractional polymorphisms (see for instance [16] or [17]). Further research is needed to study stronger definitions of homomorphism (for example [22], [7] or [9]) and see which are more adequate for the purpose of rephrasing FCSP using homomorphism problems.…”

Section: Discussion and Future Workmentioning

confidence: 99%

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Fuzzy Positive Primitive Formulas

Dellunde

2018

Modeling Decisions for Artificial Intelligence

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Can non-classical logic contribute to the analysis of complexity in computer science? In this paper, we give an step towards the solution of this open problem, taking a logical model-theoretic approach to the analysis of complexity in fuzzy constraint satisfaction. We study fuzzy positive-primitive sentences, and we present an algebraic characterization of classes axiomatized by these kind of sentences in terms of homomorphisms and finite direct products. The ultimate goal is to study the expressiveness and reasoning mechanisms of non-classical languages, with respect to constraint satisfaction problems and, in general, in modelling decision scenarios.

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Section: Discussion and Future Workmentioning

confidence: 99%

Section: Discussion and Future Workmentioning

confidence: 99%

Fuzzy Positive Primitive Formulas

Dellunde

2018

Modeling Decisions for Artificial Intelligence

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“…A complexity classification of finite-domain VCSPs for sets of cost functions taking finite (rational-only) values was established in [45]. A complexity classification of VCSPs was consequently established in [35], assuming a dichotomy for CSPs, which was proved independently in [19] and [51].…”

Section: Introductionmentioning

confidence: 99%

The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

Viola

Živný

2021

ACM Trans. Algorithms

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Convex relaxations have been instrumental in solvability of constraint satisfaction problems (CSPs), as well as in the three different generalisations of CSPs: valued CSPs, infinite-domain CSPs, and most recently promise CSPs. In this work, we extend an existing tractability result to the three generalisations of CSPs combined: We give a sufficient condition for the combined basic linear programming and affine integer programming relaxation for exact solvability of promise valued CSPs over infinite-domains. This extends a result of Brakensiek and Guruswami (SODA’20) for promise (non-valued) CSPs (on finite domains).

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“…Since the resolution of the Feder-Vardi "Dichotomy" Conjecture, classifying the complexity of CSP(Γ), for all finite Γ, between P and NP-complete [6,24], a desire has been building for a classification for QCSP(Γ). Indeed, since the classification of the Valued CSPs was reduced to that for CSPs [18], the QCSP remains the last of the older variants of the CSP to have been systematically studied but not classified. More recently, other interesting open classification questions have appeared such as that for Promise CSPs [5] and finitely-bounded, homogeneous infinite-domain CSPs [1].…”

Section: Introductionmentioning

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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The Complexity of General-Valued CSPs

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Cited by 54 publications

References 40 publications

Fuzzy Positive Primitive Formulas

Fuzzy Positive Primitive Formulas

The Combined Basic LP and Affine IP Relaxation for Promise VCSPs on Infinite Domains

QCSP monsters and the demise of the chen conjecture

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