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Strong partial clones and the time complexity of SAT problems

Peter Jonsson, Victor Lagerkvist, Gustav Nordh, Bruno Zanuttini
2017 Journal of computer and system sciences (Print)  
Hence, using 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.  ... 
doi:10.1016/j.jcss.2016.07.008 fatcat:ydgat5j4f5errokt7zrdpn2yqi

Weak bases of Boolean co-clones

Victor Lagerkvist
2014 Information Processing Letters  
To study 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  ... 
doi:10.1016/j.ipl.2014.03.011 fatcat:4rhokivpgnaxrptedshzn2emji

Bounded Bases of Strong Partial Clones

Victor Lagerkvist, Magnus Wahlstrom, Bruno Zanuttini
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  ... 
doi:10.1109/ismvl.2015.33 dblp:conf/ismvl/LagerkvistWZ15 fatcat:yptqwfjy7zg5hkofypgkixxlqy

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

Peter Jonsson, Victor Lagerkvist, Gustav Nordh, Bruno Zanuttini
2013 Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms  
Unfortunately, there is a lack 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.  ... 
doi:10.1137/1.9781611973105.92 dblp:conf/soda/LagerkvistJNZ13 fatcat:hgv3lhmyqvegbbh2cijq7yw6va

Relating the Time Complexity of Optimization Problems in Light of the Exponential-Time Hypothesis [chapter]

Peter Jonsson, Victor Lagerkvist, Johannes Schmidt, Hannes Uppman
2014 Lecture Notes in Computer Science  
(SODA 2013) show that 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.  ... 
doi:10.1007/978-3-662-44465-8_35 fatcat:giur4vixknc3bhb22dt77fkt6i

Relating the Time Complexity of Optimization Problems in Light of the Exponential-Time Hypothesis [article]

Peter Jonsson, Victor Lagerkvist, Johannes Schmidt, Hannes Uppman
2014 arXiv   pre-print
(SODA 2013) show that 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.  ... 
arXiv:1406.3247v1 fatcat:6pragflylba2rdw23shljliqee

Gap theorems for robust satisfiability: Boolean CSPs and beyond [article]

Lucy Ham
2017 arXiv   pre-print
As a consequence, we obtain a number 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.  ... 
arXiv:1610.09574v2 fatcat:5ws65mvdjzez5apnvmrooivvkm

The power of primitive positive definitions with polynomially many variables

Victor Lagerkvist, Magnus Wahlström
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.  ... 
doi:10.1093/logcom/exw005 fatcat:vdf5onpuz5a2jplxgi7zaq6say

On the Strength of Uniqueness Quantification in Primitive Positive Formulas

Victor Lagerkvist, Gustav Nordh, Michael Wagner
2019 International Symposium on Mathematical Foundations of Computer Science  
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.  ...  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.  ... 
doi:10.4230/lipics.mfcs.2019.36 dblp:conf/mfcs/LagerkvistN19 fatcat:5fmv55grojgqxkjvqtsgjcnf7m

A Dichotomy Theorem for the Inverse Satisfiability Problem

Victor Lagerkvist, Biman Roy, Marc Herbstritt
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.  ... 
doi:10.4230/lipics.fsttcs.2017.39 dblp:conf/fsttcs/LagerkvistR17 fatcat:uzdeleecbzcttpqqhg3lwdvsey

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

Victor Lagerkvist, Magnus Wahlström
2017 arXiv   pre-print
In this paper, we take an algebraic approach to 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.  ... 
arXiv:1706.05941v1 fatcat:3exvyn3oxzf4lioqs5sed6z4da

Kernelization of Constraint Satisfaction Problems: A Study Through Universal Algebra [chapter]

Victor Lagerkvist, Magnus Wahlström
2017 Lecture Notes in Computer Science  
In this paper, we take an algebraic approach to 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.  ... 
doi:10.1007/978-3-319-66158-2_11 fatcat:jbbgfbzzn5fodeao3ponbspgz4

The Constraint Satisfaction Problem: Complexity and Approximability (Dagstuhl Seminar 18231)

Martin Grohe, Venkatesan Guruswami, Stanislav Zivny, Michael Wagner
2018 Dagstuhl Reports  
CSPs constitute a very rich 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  ... 
doi:10.4230/dagrep.8.6.1 dblp:journals/dagstuhl-reports/GroheGZ18 fatcat:3bqo62ly3rgzlnh3bmkvwbuwea

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

Miguel Couceiro, Lucien Haddad, Victor Lagerqvist, Biman Roy
2017 2017 IEEE 47th International Symposium on Multiple-Valued Logic (ISMVL)  
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.  ...  ACKNOWLEDGEMENTS We thank the anonymous reviewers for several helpful suggestions on how to improve the paper.  ... 
doi:10.1109/ismvl.2017.27 dblp:conf/ismvl/CouceiroHLR17 fatcat:u3zx57o2u5f4rpvxkmq6cdwfny

Precise Upper and Lower Bounds for the Monotone Constraint Satisfaction Problem [chapter]

Victor Lagerkvist
2015 Lecture Notes in Computer Science  
We give 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.  ... 
doi:10.1007/978-3-662-48057-1_28 fatcat:hqocbudjprbjtoyyq5eqys36ne
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