Strong partial clones and the time complexity of SAT problems

Jönsson, Peter; Lagerkvist, Victor; Nordh, Gustav; Zanuttini, Bruno

doi:10.1016/j.jcss.2016.07.008

Cited by 25 publications

(57 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Put together, this implies that the set of strong partial clones containing only projections as total operations is a particularly interesting object of study, due to its relationship with NP-complete CSP problems. With these observation Jonsson et al [6] then proved that there exists a relation R = = =01…”

Section: Introductionmentioning

confidence: 92%

“…For example, the NP-complete problem 1-in-3-SAT can be seen as a CSP problem over the ternary relation R 1/3 = {(0, 0, 1), (0, 1, 0), (1, 0, 0)}. It has been proven that the polymorphisms of Γ determine the complexity of the CSP problem over Γ up to polynomial-time many-one reductions [5], while the partial polymorphisms of Γ can be used to study the complexity of the CSP problem over Γ with respect to stronger notions of reductions [6], [14]. More specifically, Jonsson et al [6] proved that if pPol(Γ) ⊆ pPol(∆) then CSP(∆) is solvable in O(c n ) time whenever CSP(Γ) is solvable in O(c n ) time (if Γ and ∆ are both finite sets of relations).…”

Section: Introductionmentioning

confidence: 99%

“…) is the largest set of Boolean partial operations which does not contain any total operations (except the projections) [6], [14]. Jonsson et al [6] then conjectured that the set of strong partial clones between pPol(R 1/3 ) and pPol(R = = =01…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

On the Interval of Boolean Strong Partial Clones Containing Only Projections as Total Operations

Couceiro

Haddad

Lagerqvist³

et al. 2017

2017 IEEE 47th International Symposium on Multiple-Valued Logic (ISMVL)

View full text Add to dashboard Cite

A strong partial clone is a set of partial operations closed under composition and containing all partial projections. Let X be the set of all Boolean strong partial clones whose total operations are the projections. This set is of practical interest since it induces a partial order on the complexity of NP-complete constraint satisfaction problems. In this paper we study X from the algebraic point of view, and prove that there exists two intervals in X , corresponding to natural constraint satisfaction problems, such that one is at least countably infinite and the other has the cardinality of the continuum.

show abstract

Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the Interval of Boolean Strong Partial Clones Containing Only Projections as Total Operations

Couceiro

Haddad

Lagerqvist³

et al. 2017

2017 IEEE 47th International Symposium on Multiple-Valued Logic (ISMVL)

View full text Add to dashboard Cite

show abstract

“…For example, there are numerous dichotomy results for the complexity of parameterized SAT(Γ) and CSP(Γ) problems, both for so-called FPT algorithms and for kernelization [30,31,32,36], and in each of the cases listed, a dichotomy is given which is equivalent to requiring a finite list of partial polymorphisms of Γ. Similarly, Jonsson et al [29] showed that the exact running times of NP-hard SAT(Γ) and CSP(Γ) problems, in terms of the number of variables n, are characterized by the partial polymorphisms of the constraint language Γ. Recently, this approach was also applied in a research programme of classifying NP-hard SAT(Γ) problems admitting exponentially improved upper bounds [34].…”

Section: The Algebraic Approach In Parameterized and Fine-grained Commentioning

confidence: 99%

Sparsification of SAT and CSP Problems via Tractable Extensions

Lagerkvist

Wahlström

2020

ACM Trans. Comput. Theory

Self Cite

View full text Add to dashboard Cite

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.

show abstract

“…Using this function we may state the following theorem. [21]) Let Γ 1 and Γ 2 be two finite constraint languages over D.…”

Section: Algebra In Cspmentioning

confidence: 99%

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

Roy

2020

Linköping Studies in Science and Technology. Dissertations

View full text Add to dashboard Cite

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.

show abstract

Strong partial clones and the time complexity of SAT problems

Cited by 25 publications

References 31 publications

On the Interval of Boolean Strong Partial Clones Containing Only Projections as Total Operations

On the Interval of Boolean Strong Partial Clones Containing Only Projections as Total Operations

Sparsification of SAT and CSP Problems via Tractable Extensions

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

Contact Info

Product

Resources

About