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Strong partial clones and the time complexity of SAT problems

2017
*
Journal of computer and system sciences (Print)
*

Hence, using

doi:10.1016/j.jcss.2016.07.008
fatcat:ydgat5j4f5errokt7zrdpn2yqi
*clone*theory, we obtain a solid understanding*of**the**complexity**of**SAT*(S) with CV-*and*LV-reductions. ... Improving exact exponential-*time*algorithms for NP-complete*problems*is an expanding research area. Unfortunately, general methods for comparing*the**complexity**of*such*problems*is sorely lacking. ... Acknowledgments We thank Magnus Wahlström for helpful discussions on*the*topic*of*this article,*and*in particular for suggesting*the*proof strategy in Lemma 17. ...##
###
Weak bases of Boolean co-clones

2014
*
Information Processing Letters
*

To study

doi:10.1016/j.ipl.2014.03.011
fatcat:4rhokivpgnaxrptedshzn2emji
*the**complexity**of*CSP(S) we borrow methods from universal algebra. In particular, we consider algebras*of**partial*functions, called*strong**partial**clones*. ... Using this classification we prove that a*strong**partial**clone*is*of*infinite order if (1)*the*total functions in*the**strong**partial**clone*are essentially unary*and*(2)*the*corresponding constraint language ... Second, we show that both MONO-TONE 1-IN-κ-*SAT**and*κ-*SAT*correspond to different*strong**partial**clones*for every κ*and*also that*the**strong**partial**clones*corresponding to MONO-TONE 1-IN-(κ + 1)-*SAT**and*...##
###
Bounded Bases of Strong Partial Clones

2015
*
2015 IEEE International Symposium on Multiple-Valued Logic
*

*Partial*

*clone*theory has successfully been applied to study

*the*

*complexity*

*of*

*the*constraint satisfaction

*problem*parameterized by a set

*of*relations (CSP(Γ)). ... Determining bounded bases for finite languages provides a complete characterization

*of*their

*partial*polymorphisms

*and*we provide such bases for k-

*SAT*

*and*1-in-k-

*SAT*. ...

*Strong*

*partial*

*clones*are therefore useful when comparing

*the*

*time*

*complexity*between NP-hard

*problems*,

*and*have for instance been used to find

*the*computationally easiest NP-hard CSP

*problem*for finite ...

##
###
Complexity of SAT Problems, Clone Theory and the Exponential Time Hypothesis
[chapter]

2013
*
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
*

Unfortunately, there is a lack

doi:10.1137/1.9781611973105.92
dblp:conf/soda/LagerkvistJNZ13
fatcat:hgv3lhmyqvegbbh2cijq7yw6va
*of*general methods for studying*and*comparing*the**time**complexity**of*algorithms for such*problems*. ... We show that there is a certain*partial*order on*the*NP-complete*SAT**problems*with a close connection to their worst-case*time**complexities*; if a*problem**SAT*(S) is below a*problem**SAT*(S ) in this*partial*... Acknowledgments We thank Magnus Wahlström for helpful discussions on*the*topic*of*this paper. ...##
###
Relating the Time Complexity of Optimization Problems in Light of the Exponential-Time Hypothesis
[chapter]

2014
*
Lecture Notes in Computer Science
*

(SODA 2013) show that

doi:10.1007/978-3-662-44465-8_35
fatcat:giur4vixknc3bhb22dt77fkt6i
*the**time**complexity**of**the*parameterized*SAT*(·)*problem*correlates to*the*lattice*of**strong**partial**clones*. ... With*the*help*of*these languages we relate MAX-ONES*and*VCSP to*the*exponential*time*hypothesis in several different ways. ⋆ This is an extended version*of*Relating*the**Time**Complexity*... Obviously sets*of**the*form pPol(Γ) always form*strong**partial**clones**and*again we have a Galois connection between*clones**and*co-*clones*. ...##
###
Relating the Time Complexity of Optimization Problems in Light of the Exponential-Time Hypothesis
[article]

2014
*
arXiv
*
pre-print

(SODA 2013) show that

arXiv:1406.3247v1
fatcat:6pragflylba2rdw23shljliqee
*the**time**complexity**of**the*parameterized*SAT*(·)*problem*correlates to*the*lattice*of**strong**partial**clones*. ... With*the*help*of*these languages we relate MaxOnes*and*VCSP to*the*exponential*time*hypothesis in several different ways. ... Obviously sets*of**the*form pPol(Γ) always form*strong**partial**clones**and*again we have a Galois connection between*clones**and*co-*clones*. ...##
###
Gap theorems for robust satisfiability: Boolean CSPs and beyond
[article]

2017
*
arXiv
*
pre-print

As a consequence, we obtain a number

arXiv:1610.09574v2
fatcat:5ws65mvdjzez5apnvmrooivvkm
*of*dichotomies for*the**complexity**of*specific variants*of**the*constraint satisfaction*problem*: all are either polynomial-*time*tractable or NP-complete. ... We establish a Gap Trichotomy Theorem for a family*of*constraint*problem*variants, completely classifying*the**complexity**of*possible NP-hard gaps in*the*case*of*Boolean domains. ...*The*author is indebted to Marcel Jackson for many useful suggestions*and*discussions, as well as her PhD supervisors Brian Davey*and*Tomasz Kowalski, for their guidance*and*feedback. ...##
###
The power of primitive positive definitions with polynomially many variables

2016
*
Journal of Logic and Computation
*

*The*latter do however have limited expressivity,

*and*

*the*corresponding lattice

*of*

*strong*

*partial*

*clones*is

*of*uncountably innite cardinality even for

*the*Boolean domain. ... We prove that if Γ is a nite set

*of*relations over an arbitrary nite domain such that

*the*

*clone*corresponding to Γ is essentially unary, then

*the*

*strong*

*partial*

*clone*corresponding to Γ is

*of*innite order ... Acknowledgements

*The*authors are grateful toward Peter Jonsson, Karsten Schölzel

*and*Bruno Zanuttini, for helpful comments

*and*suggestions. ...

##
###
On the Strength of Uniqueness Quantification in Primitive Positive Formulas

2019
*
International Symposium on Mathematical Foundations of Computer Science
*

Studying these

doi:10.4230/lipics.mfcs.2019.36
dblp:conf/mfcs/LagerkvistN19
fatcat:5fmv55grojgqxkjvqtsgjcnf7m
*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. ... 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*. ... Acknowledgements We thank Andrei Bulatov for helpful discussions concerning*the*topic*of**the*paper,*and**the*anonymous reviewers for their constructive feedback. ...##
###
A Dichotomy Theorem for the Inverse Satisfiability Problem

2018
*
Foundations of Software Technology and Theoretical Computer Science
*

*The*inverse satisfiability

*problem*over a set

*of*Boolean relations Γ (Inv-

*SAT*(Γ)) is

*the*computational decision

*problem*

*of*, given a relation R, deciding whether there exists a

*SAT*(Γ) instance with R as ... Very few

*of*

*the*techniques used by Kavvadias

*and*Sideri are applicable

*and*we have to turn to more recently developed algebraic approaches based on

*partial*polymorphisms. ... We thank

*the*anonymous reviewers for several helpful comments. ...

##
###
Kernelization of Constraint Satisfaction Problems: A Study through Universal Algebra
[article]

2017
*
arXiv
*
pre-print

In this paper, we take an algebraic approach to

arXiv:1706.05941v1
fatcat:3exvyn3oxzf4lioqs5sed6z4da
*the**problem**of*characterizing*the*kernelization limits*of*NP-hard*SAT**and*CSP*problems*, parameterized by*the*number*of*variables. ... For*the*Boolean satisfiability*problem*(*SAT*),*and**the*constraint satisfaction*problem*(CSP), there exist many results concerning upper*and*lower bounds for kernelizability*of*specific*problems*, but it ... [22] 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*Γ. ...##
###
Kernelization of Constraint Satisfaction Problems: A Study Through Universal Algebra
[chapter]

2017
*
Lecture Notes in Computer Science
*

In this paper, we take an algebraic approach to

doi:10.1007/978-3-319-66158-2_11
fatcat:jbbgfbzzn5fodeao3ponbspgz4
*the**problem**of*characterizing*the*kernelization limits*of*NP-hard*SAT**and*CSP*problems*, parameterized by*the*number*of*variables. ... For*the*Boolean satisfiability*problem*(*SAT*),*and**the*constraint satisfaction*problem*(CSP), there exist many results concerning upper*and*lower bounds for kernelizability*of*specific*problems*, but it ... [22] 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*Γ. ...##
###
The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231)

2018
*
Dagstuhl Reports
*

CSPs constitute a very rich

doi:10.4230/dagrep.8.6.1
dblp:journals/dagstuhl-reports/GroheGZ18
fatcat:3bqo62ly3rgzlnh3bmkvwbuwea
*and*yet sufficiently manageable class*of**problems*to give a good perspective on general computational phenomena. ... Constraint satisfaction has always played a central role in computational*complexity*theory; appropriate versions*of*CSPs are classical complete*problems*for most standard*complexity*classes. ... -*The*Constraint Satisfaction*Problem*:*Complexity**and*Approximability From Weak to*Strong*LP Gaps for all CSPs We study*the*approximability*of*constraint satisfaction*problems*(CSPs) by linear programming ...##
###
On the Interval of Boolean Strong Partial Clones Containing Only Projections as Total Operations

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

A

doi:10.1109/ismvl.2017.27
dblp:conf/ismvl/CouceiroHLR17
fatcat:u3zx57o2u5f4rpvxkmq6cdwfny
*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. ... ACKNOWLEDGEMENTS We thank*the*anonymous reviewers for several helpful suggestions on how to improve*the*paper. ...##
###
Precise Upper and Lower Bounds for the Monotone Constraint Satisfaction Problem
[chapter]

2015
*
Lecture Notes in Computer Science
*

We give

doi:10.1007/978-3-662-48057-1_28
fatcat:hqocbudjprbjtoyyq5eqys36ne
*strong*lower bounds*and*prove that MCSP(Γ ), for any constraint language Γ over any finite domain, is solvable in O(|D | n )*time*, where D is*the*domain*of**the*core*of*Γ , but not solvable in O ... We prove that*the**complexity**of**the*NP-complete MCSP(Γ )*problems*on a given finite domain D falls into exactly |D| − 1 cases*and*ranges from O(2 n ) to O(|D| n ). ... Acknowledgments*The*author is grateful towards Peter Jonsson for several helpful discussions regarding*the*content*of*this paper,*and*to*the*anonymous reviewers for many suggestions for improvement. ...
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