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Promise Constraint Satisfaction: Algebraic Structure and a Symmetric Boolean Dichotomy [article]

Joshua Brakensiek, Venkatesan Guruswami
2021 arXiv   pre-print
A classic result due to Schaefer (1978) classifies all constraint satisfaction problems (CSPs) over the Boolean domain as being either in 𝖯 or 𝖭𝖯-hard.  ...  Such insights call us to question the existence of a general dichotomy for Boolean PCSPs.  ...  In this work we are interested in a potential complexity dichotomy for promise constraint satisfaction problems (PCSPs).  ... 
arXiv:1704.01937v2 fatcat:iwsddj3pvnedrotkrksgl5l4du

Finitely Tractable Promise Constraint Satisfaction Problems [article]

Kristina Asimi, Libor Barto
2020 arXiv   pre-print
cases in the dichotomy theorem for symmetric Boolean PCSPs allowing negations by Brakensiek and Guruswami [SODA'18].  ...  The Promise Constraint Satisfaction Problem (PCSP) is a generalization of the Constraint Satisfaction Problem (CSP) that includes approximation variants of satisfiability and graph coloring problems.  ...  Is it so for all symmetric Boolean templates? For general Boolean templates, the answer is "No": [12] presents an example that requires a three-element C.  ... 
arXiv:2010.04618v1 fatcat:zezjn7ux6nappp3op6fnu6geqy

Algebraic approach to promise constraint satisfaction [article]

Libor Barto, Jakub Bulín, Andrei Krokhin, Jakub Opršal
2019 arXiv   pre-print
The complexity and approximability of the constraint satisfaction problem (CSP) has been actively studied over the last 20 years.  ...  We introduce a new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem, and show that every PCSP with a fixed constraint language is equivalent to a problem of this  ...  Acknowledgements We would like to thank Libor Barto, Mirek Olšák, Alex Kazda, Josh Brakensiek, Venkat Guruswami, Erkko Lehtonen, and Marcello Mamino for valuable discussions.  ... 
arXiv:1811.00970v3 fatcat:r2pgvc74ffcxzec6fqdkwvznm4

Symmetric Promise Constraint Satisfaction Problems: Beyond the Boolean Case [article]

Libor Barto, Diego Battistelli, Kevin M. Berg
2020 arXiv   pre-print
The Promise Constraint Satisfaction Problem (PCSP) is a recently introduced vast generalization of the Constraint Satisfaction Problem (CSP).  ...  by a ternary symmetric relation.  ...  Introduction The Constraint Satisfaction Problem (CSP) over a finite relational structure A (also called a template), denoted CSP(A), can be defined as a homomorphism problem with a fixed target structure  ... 
arXiv:2010.04623v1 fatcat:yrmqlyq36rbcpn7vsuxebfxgyy

Topology and adjunction in promise constraint satisfaction [article]

Andrei Krokhin, Jakub Opršal, Marcin Wrochna, Stanislav Živný
2020 arXiv   pre-print
This problem naturally generalises to promise graph homomorphism and further to promise constraint satisfaction problems. Complexity analysis of all these problems is notoriously difficult.  ...  We apply these techniques, together with the previously introduced algebraic approach, to obtain new NP-hardness results for a significant class of approximate graph colouring and promise graph homomorphism  ...  We remark that the above proposition and its proof generalizes to arbitrary Hom complexes Hom(H, G) with the action of Aut(H), more precisely, the complex Hom prod (H, G) (as de ned in [Koz08, De nition  ... 
arXiv:2003.11351v2 fatcat:w45dasesbbgtbbfhhbboipadry

The Power of the Combined Basic Linear Programming and Affine Relaxation for Promise Constraint Satisfaction Problems

Joshua Brakensiek, Venkatesan Guruswami, Marcin Wrochna, Stanislav Živný
2020 SIAM journal on computing (Print)  
rich algebraic structure of a clone (c.f., [4] ).  ...  In the field of constraint satisfaction problems (CSPs), promise CSPs are an exciting new direction of study.  ...  We thank Libor Barto, Andrei Krokhin, and Jakub Opr\v sal for useful comments and encouragement. We also thank anonymous reviewers for many helpful comments.  ... 
doi:10.1137/20m1312745 fatcat:hd7moxv3crbkbeulczbttxv7am

All or nothing: toward a promise problem dichotomy for constraint problems [article]

Lucy Ham, Marcel Jackson
2017 arXiv   pre-print
We show that intractability of the constraint satisfaction problem CSP(R) can, in all known cases, be replaced by an infinite hierarchy of intractable promise problems of increasingly disparate promise  ...  A finite constraint language R is a finite set of relations over some finite domain A.  ...  The promise conditions include satisfaction as a special case, and complement the promise condition on NO instances provided by the PCP Theorem [3] (at least ε proportion of the constraints must fail  ... 
arXiv:1611.00886v3 fatcat:p2z7sjjmgzdyrgzt7xmtl2sdne

All or Nothing: Toward a Promise Problem Dichotomy for Constraint Problems [chapter]

Lucy Ham, Marcel Jackson
2017 Lecture Notes in Computer Science  
We show that intractability of the constraint satisfaction problem CSP(R) can, in all known cases, be replaced by an infinite hierarchy of intractable promise problems of increasingly disparate promise  ...  A finite constraint language R is a finite set of relations over some finite domain A.  ...  The promise conditions include satisfaction as a special case, and complement the promise condition on NO instances provided by the PCP Theorem [3] (at least ε proportion of the constraints must fail  ... 
doi:10.1007/978-3-319-66158-2_10 fatcat:4yjuuspi45aldejfbo527cd4fq

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

Martin Grohe, Venkatesan Guruswami, Stanislav Zivny, Michael Wagner
2018 Dagstuhl Reports  
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.  ...  CSPs constitute a very rich and yet sufficiently manageable class of problems to give a good perspective on general computational phenomena.  ...  -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

Constraint Satisfaction Problems over Finite Structures [article]

Libor Barto and William DeMeo and Antoine Mottet
2021 arXiv   pre-print
We initiate a systematic study of the computational complexity of the Constraint Satisfaction Problem (CSP) over finite structures that may contain both relations and operations.  ...  We give some sufficient and some necessary conditions for a finite algebra to have this property.  ...  Constraint Satisfaction Problems.  ... 
arXiv:2010.04958v2 fatcat:osswdtu2zrbatjkodmvw4dyg74

Universal algebra and hardness results for constraint satisfaction problems

Benoît Larose, Pascal Tesson
2009 Theoretical Computer Science  
We present algebraic conditions on constraint languages Γ that ensure the hardness of the constraint satisfaction problem CSP(Γ ) for complexity classes L, NL, P, NP and Mod p L.  ...  can show that the complexity of CSP(Γ ) depends on the algebraic structure of A(Γ ).  ...  We are also grateful to the organizers of the Dagstuhl Seminar on the Complexity of Constraints where much of this work was completed.  ... 
doi:10.1016/j.tcs.2008.12.048 fatcat:qm6drmlez5cy3iqz6xv5cvjazi

Constraint Satisfaction Problems with Infinite Templates [chapter]

Manuel Bodirsky
2008 Lecture Notes in Computer Science  
We close with a series of open problems and promising directions of future research.  ...  We then give an introduction to the universal-algebraic approach to infinite-domain constraint satisfaction, and discuss how cores, polymorphism clones, and pseudovarieties can be used to study the computational  ...  I would like to thank the referee for his critical and helpful comments, all my CSP co-authors for the cooperation: Hubie Chen, Victor Dalmau, Jan Kára, Martin Kutz, Jaroslav Nešetřil, Michael Pinsker,  ... 
doi:10.1007/978-3-540-92800-3_8 fatcat:tzhsee2mkjhy5bk7i5jmpiraee

Finitely Tractable Promise Constraint Satisfaction Problems

Kristina Asimi, Libor Barto, Filippo Bonchi, Simon J. Puglisi
2021
cases in the dichotomy theorem for symmetric Boolean PCSPs allowing negations by Brakensiek and Guruswami [SODA'18].  ...  The Promise Constraint Satisfaction Problem (PCSP) is a generalization of the Constraint Satisfaction Problem (CSP) that includes approximation variants of satisfiability and graph coloring problems.  ...  A doubly cyclic function t : A p 2 → B is b-bounded if Finitely Tractable Promise Constraint Satisfaction Problems The promised necessary condition for finite tractability is now a simple consequence  ... 
doi:10.4230/lipics.mfcs.2021.11 fatcat:drl5nugo7bb6jacbrvbixmbrea

Complexity Classification in Infinite-Domain Constraint Satisfaction [article]

Manuel Bodirsky
2019 arXiv   pre-print
A constraint satisfaction problem (CSP) is a computational problem where the input consists of a finite set of variables and a finite set of constraints, and where the task is to decide whether there exists  ...  We also study the limits of complexity classification, and present classes of computational problems provably do not exhibit a complexity dichotomy into hard and easy problems.  ...  The link between the set constraint satisfaction problems over 2 N mentioned in Section 1.5 and the atomless Boolean algebra is the following. Proposition 4.3.2. Let C be a set constraint language.  ... 
arXiv:1201.0856v10 fatcat:t32gozkmqrauze2gzqn2ze5kaa

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

Victor Lagerkvist, Magnus Wahlström
2017 arXiv   pre-print
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  ...  We also study extensions of this towards SAT and CSP problems with kernels with O(n^c) constraints, c>1, based on embeddings into CSP problems preserved by a k-edge operation, k > c.  ...  Polymorphisms and Partial Polymorphisms In this section we define the link between constraint languages and algebras that was promised in Section 1.  ... 
arXiv:1706.05941v1 fatcat:3exvyn3oxzf4lioqs5sed6z4da
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