Strong Partial Clones and the Complexity of Constraint Satisfaction Problems : Limitations and Applications

Lagerkvist, Victor

doi:10.3384/diss.diva-122827

Cited by 5 publications

(7 citation statements)

References 53 publications

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“…The latter resolves an open question in Chen [11]. Schnoor and Schnoor [30] proved the existence of weak bases for every Boolean clone, and Lagerkvist gave a simplified list of weak bases for all Boolean clones [26]. We completely characterize the strong partial clones of weak bases according to set inclusions.…”

Section: Paper 4: a Dichotomy Theorem For The Inverse Satisfiabilmentioning

confidence: 65%

Applications of Partial Polymorphisms in (Fine-Grained) Complexity of Constraint Satisfaction Problems

Roy

2020

Linköping Studies in Science and Technology. Dissertations

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In this thesis we study the worst-case complexity of constraint satisfaction problems and some of its variants. We use methods from universal algebra: in particular, algebras of total functions and partial functions that are respectively known as clones and strong partial clones. The constraint satisfaction problem parameterized by a set of relations Γ (CSP(Γ)) is the following problem: given a set of variables restricted by a set of constraints based on the relations Γ, is there an assignment to the variables that satisfies all constraints? We refer to the set Γ as a constraint language. The inverse CSP problem over Γ (I-CSP(Γ)) asks the opposite: given a relation R, does there exist a CSP(Γ) instance with R as its set of models? When Γ is a Boolean language, then we use the term SAT(Γ) instead of CSP(Γ) and I-SAT(Γ) instead of I-CSP(Γ). Fine-grained complexity is an approach in which we zoom inside a complexity class and classify the problems in it based on their worst-case time complexities. We start by investigating the fine-grained complexity of NP-complete CSP(Γ) problems. An NP-complete CSP(Γ) problem is said to be easier than an NP-complete CSP(∆) problem if the worst-case time complexity of CSP(Γ) is not higher than 5. Victor Lagerkvist and Biman Roy. The Inclusion Structure of Boolean Weak Bases.

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Section: Paper 4: a Dichotomy Theorem For The Inverse Satisfiabilmentioning

confidence: 65%

Applications of Partial Polymorphisms in (Fine-Grained) Complexity of Constraint Satisfaction Problems

Roy

2020

Linköping Studies in Science and Technology. Dissertations

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“…See Table 2 for a list of weak bases for the Boolean co-clones of interest in this paper [20,21]. Here, and in the sequel, we use the co-clone terminology developed by Reith & Wagner [27] and Böhler et al [5], where a Boolean co-clone Inv(C) is typically written as IC.…”

Section: Weak and Plain Bases Of Co-clonesmentioning

confidence: 99%

On the Strength of Uniqueness Quantification in Primitive Positive Formulas

Lagerkvist,

Nordh

2019

Preprint

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Uniqueness quantification (∃!) is a quantifier in first-order logic where one requires that exactly one element exists satisfying a given property. In this paper we investigate the strength of uniqueness quantification when it is used in place of existential quantification in conjunctive formulas over a given set of relations Γ, so-called primitive positive definitions (pp-definitions). We fully classify the Boolean sets of relations where uniqueness quantification has the same strength as existential quantification in pp-definitions and give several results valid for arbitrary finite domains. We also consider applications of ∃!-quantified pp-definitions in computer science, which can be used to study the computational complexity of problems where the number of solutions is important. Using our classification we give a new and simplified proof of the trichotomy theorem for the unique satisfiability problem, and prove a general result for the unique constraint satisfaction problem. Studying these problems in a more rigorous framework also turns out to be advantageous in the context of lower bounds, and we relate the complexity of these problems to the exponential-time hypothesis.

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“…This is a natural restriction which holds for many well-studied constraint language, e.g., the languges corresponding to k-SAT, 1--k-SAT and the roots of bounded-degree polynomials are all sign-symmetric. Furthermore, it is known that the expressive power of a sign-symmetric constraint language is characterised by a restricted kind of partial polymorphism which we refer to as pSDIoperations (for partial, self-dual and idempotent) [34,38]. Thus, the restriction to sign-symmetric languages corresponds directly to a restriction on the algebraic level.…”

Section: Sat(γ) Admits An Improved Algorithm In the Oracle Model If S...mentioning

confidence: 99%

“…In plain language, these results show that finite constraint languages result in complex partial polymorphisms, and that simple partial polymorphisms result in complex constraint languages. A previous attempt at grappling with this difficulty provided closure operators that generate pPol(Γ) for a finite Γ from a finite basis [34], but this intrinsically uses that Γ is finite, and is not applicable in the current paper.…”

Section: Related Workmentioning

confidence: 99%

Which NP-Hard SAT and CSP Problems Admit Exponentially Improved Algorithms?

Lagerkvist,

Wahlström

2018

Preprint

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We study the complexity of SAT(Γ) problems for potentially infinite languages Γ closed under variable negation, which we refer to as sign-symmetric languages Γ. Via an algebraic connection, this reduces to the study of restricted partial polymorphisms we refer to as pSDI-operations (for partial, self-dual and idempotent), under which the language Γ is invariant. First, we focus on the language classes themselves. We classify the structure of the least restrictive pSDI-operations, corresponding to the most powerful languages Γ, and find that these operations can be divided into levels, corresponding to a rough notion of difficulty, where every level k has an easiest language class, containing the language for (k − 1)-SAT, and a hardest language class, containing (among other things) constraints encoded as roots of multivariate polynomials of degree (k − 1). Particular classes in each level correspond to the natural partially defined versions of previously studied total algebraic invariants. In particular, the easiest class on level k ≥ 3 corresponds to the partial k-ary near-unanimity (k-NU) operation, and a larger class corresponds to the partial k-edge operation. The largest class at each level corresponds to a partial operation u k we call k-universal. Furthermore, every sign-symmetric language Γ not preserved by u k implements all k-clauses, hence SAT(Γ) is at least as hard as k-SAT; and if Γ is not preserved by u k for any k, then SAT(Γ) is trivially SETH-hard (i.e., takes time O * (2 n ) under SETH).Second, we consider implications of this for the complexity of SAT(Γ). We find that particular classes in the hierarchy correspond to previously known algorithmic strategies. In particular, languages preseved by the partial 2-edge operation can be solved via S S -style meet in the middle, and languages preserved by the partial 3-NU operation can be solved via fast matrix multiplication. These results also hold for the correspondning non-Boolean CSP problems. We also find that symmetric 3-edge languages reduce to finding a monochromatic triangle in an edge-coloured graph, which can be done using algorithms for sparse matrix multiplication; and if the sunflower conjecture holds for sunflowers with k petals, then the partial k-NU language has an improved algorithm via Schöning-style local search.Complementing this, we show a lower bound, showing that for every level k there is a constant c k such that for every partial operation p on level k, the problem SAT(Γ) with Γ = Inv(p) cannot be solved faster than O * (c n k ) unless SETH fails. In particular, when Γ = Inv(2-edge), this gives us the first NP-hard SAT problem which simultaneously has non-trivial upper and lower bounds on the running time, assuming SETH. Finally, we note a possible conjecture: It is consistent with our present knowledge that SAT(Γ) admits an improved algorithm if and only if Γ is preserved by u k for some constant k. However, to show this in the positive poses some significant difficulty.

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Strong Partial Clones and the Complexity of Constraint Satisfaction Problems : Limitations and Applications

Cited by 5 publications

References 53 publications

Applications of Partial Polymorphisms in (Fine-Grained) Complexity of Constraint Satisfaction Problems

Applications of Partial Polymorphisms in (Fine-Grained) Complexity of Constraint Satisfaction Problems

On the Strength of Uniqueness Quantification in Primitive Positive Formulas

Which NP-Hard SAT and CSP Problems Admit Exponentially Improved Algorithms?

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