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The expressibility of functions on the boolean domain, with applications to counting CSPs

Andrei A. Bulatov, Martin Dyer, Leslie Ann Goldberg, Mark Jerrum, Colin Mcquillan
2013 Journal of the ACM  
One of these clones is the collection of log-supermodular (lsm) functions, which turns out to play a significant role in classifying counting CSPs.  ...  Post's lattice gives a complete classification of all Boolean relational clones, and this has been used to classify the computational difficulty of CSPs.  ...  The expressibility of functions on the Boolean domain, with applications to Counting CSPs A:19 IBF IR 0 IR 1 IR 2 IM IM 0 IM 1 IM 2 ID IL IE IV IN II ID 1 ID 2 IL 1 IL 0  ... 
doi:10.1145/2528401 fatcat:5kodphtjxjcobgginnygpcl7py

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

Boolean Constraint Satisfaction Problems: When Does Post's Lattice Help? [chapter]

Nadia Creignou, Heribert Vollmer
2008 Lecture Notes in Computer Science  
Pol(Γ) is the set of all polymorphisms of Γ, i.e., the set of all Boolean functions that preserve every relation in Γ.  ...  Then R expresses equality. Analogous argument with NAND m for Γ ⊆ Inv(S 12 ). Non-FO CSPs are logspace-hard: Talk by Benoît Larose -Obtain fine classification for Boolean counting problem.  ... 
doi:10.1007/978-3-540-92800-3_2 fatcat:5ahjgjte5vfalii7u7rre6kdkq

Counting Problems and Clones of Functions

Andrei A. Bulatov
2009 2009 39th International Symposium on Multiple-Valued Logic  
The Counting Constraint Satisfaction Problem (#CSP for short) provides a powerful yet convenient formalism to express counting problems.  ...  In some cases, such as exact counting of solutions to CSPs, this conncetion allows one to classify completely the complexity of such problems and develop new solution algorithms.  ...  The set of all tuples from D n on which f is defined is called the domain of f and denoted by dom(f ).  ... 
doi:10.1109/ismvl.2009.47 dblp:conf/ismvl/Bulatov09 fatcat:4xtyqeumsvbr3lcnfu4ukzfrsy

An approximation trichotomy for Boolean #CSP

Martin Dyer, Leslie Ann Goldberg, Mark Jerrum
2010 Journal of computer and system sciences (Print)  
We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance.  ...  This means that the problem of approximately counting satisfying assignments of such a CSP instance is equivalent in complexity to several other known counting problems, including the problem of approximately  ...  In this paper we build on previous work on the complexity of approximate counting to identify a trichotomy in the complexity of approximate counting for Boolean #CSP.  ... 
doi:10.1016/j.jcss.2009.08.003 fatcat:sod7lmaym5eg7kz34bmg4hpk6e

Identifying Efficiently Solvable Cases of Max CSP [chapter]

David Cohen, Martin Cooper, Peter Jeavons, Andrei Krokhin
2004 Lecture Notes in Computer Science  
Finally, we show that the equality constraint over a non-Boolean domain is non-supermodular, and, when combined with some simple unary constraints, gives rise to cases of Max CSP which are hard even to  ...  Here we obtain the first examples of general families of efficiently solvable cases of Max CSP for arbitrary finite domains, by considering supermodular functions on finite lattices.  ...  In Sections 3 and 4, we give two different generalizations for the (unique) non-trivial tractable case of Boolean Max CSP: one to general supermodular constraints on restricted types of ordered domains  ... 
doi:10.1007/978-3-540-24749-4_14 fatcat:k2fq6aljyzan3e2hn376u6zley

The Complexity of Problems for Quantified Constraints

Michael Bauland, Elmar Böhler, Nadia Creignou, Steffen Reith, Henning Schnoor, Heribert Vollmer
2009 Theory of Computing Systems  
Generalizing these results to non Boolean domains, we obtain a number of hardnes results for quantified constraints over arbitrary finite universes.  ...  In this paper we will look at restricted versions of the evaluation problem, the model checking problem, the equivalence problem, and the counting problem for quantified propositional formulas, both with  ...  We are grateful to Miki Hermann for many helpful comments on different topics of this paper, and to Arnaud Durand for suggesting the last open question in our conclusion.  ... 
doi:10.1007/s00224-009-9194-6 fatcat:if5enr6ltvhebeo4vimq32qzsi

An approximation trichotomy for Boolean #CSP [article]

Martin Dyer, Leslie Ann Goldberg, Mark Jerrum
2009 arXiv   pre-print
We give a trichotomy theorem for the complexity of approximately counting the number of satisfying assignments of a Boolean CSP instance.  ...  This means that the problem of approximately counting satisfying assignments of such a CSP instance is equivalent in complexity to several other known counting problems, including the problem of approximately  ...  We have recently [10] extended Creignou and Hermann's dichotomy to the domain of weighted Boolean #CSP giving an effective dichotomy between FP and FP #P for the problem of computing the partition function  ... 
arXiv:0710.4272v2 fatcat:dxdbyt357jg4refsyexdkzf3fi

Optimal Polynomial-Time Compression for Boolean Max CSP

Bart M. P. Jansen, Michał Włodarczyk, Peter Sanders, Grzegorz Herman, Fabrizio Grandoni
2020 European Symposium on Algorithms  
In the Boolean maximum constraint satisfaction problem - Max CSP(Γ) - one is given a collection of weighted applications of constraints from a finite constraint language Γ, over a common set of variables  ...  , and the goal is to assign Boolean values to the variables so that the total weight of satisfied constraints is maximized.  ...  On the exponential-time front, we have shown that Max d-CNF-SAT is as hard as any Max CSP of degree d as long as negative weights are allowed.  ... 
doi:10.4230/lipics.esa.2020.63 dblp:conf/esa/Jansen020 fatcat:vdmlgng22nebvltgqwmnfgmei4

The complexity of weighted Boolean #CSP with mixed signs

Andrei Bulatov, Martin Dyer, Leslie Ann Goldberg, Markus Jalsenius, David Richerby
2009 Theoretical Computer Science  
Such a problem is parameterized by a set Γ of rational-valued functions, which generalize constraints. Each function assigns a weight to every assignment to a set of Boolean variables.  ...  We give a complexity dichotomy for the problem of computing the partition function of a weighted Boolean constraint satisfaction problem.  ...  In this paper we are interested only in the Boolean case, where q = 2. A constraint language Γ with domain {0, 1, . . . , q − 1} is a set of relations on {0, 1, . . . , q − 1}.  ... 
doi:10.1016/j.tcs.2009.06.003 fatcat:kghgqstmrff77blhqgijn6kkm4

The Complexity of Weighted Boolean #CSP with Mixed Signs [article]

Andrei Bulatov, Martin Dyer, Leslie Ann Goldberg, Markus Jalsenius and David Richerby
2009 arXiv   pre-print
Such a problem is parameterized by a set of rational-valued functions, which generalize constraints. Each function assigns a weight to every assignment to a set of Boolean variables.  ...  We give a complexity dichotomy for the problem of computing the partition function of a weighted Boolean constraint satisfaction problem.  ...  In this paper we are interested only in the Boolean case, where q = 2. A constraint language Γ with domain {0, 1, . . . , q − 1} is a set of relations on {0, 1, . . . , q − 1}.  ... 
arXiv:0812.4171v2 fatcat:muim3mn4w5dfvdoi5nhqldreeu

Tractability in constraint satisfaction problems: a survey

Clément Carbonnel, Martin C. Cooper
2015 Constraints  
Acknowledgments We are grateful to Peter Jeavons and StanislavŽivný for their detailed comments on a first draft of this paper, and to the reviewers for their constructive comments.  ...  Complexity dichotomies are known for different versions of the problem of computing the partition function of a weighted CSP, in the case of the Boolean domain [80, 29] .  ...  Solutions to I are in one-to-one correspondence with the n-cliques of the microstructure of I and with the size-n independent sets of the microstructure complement of I.  ... 
doi:10.1007/s10601-015-9198-6 fatcat:fl7kxmceh5bqzpyd2c37rtohii

Computational Complexity of Constraint Satisfaction [chapter]

Heribert Vollmer
2007 Lecture Notes in Computer Science  
The input to a constraint satisfaction problem (CSP) consists of a set of variables, each with a domain, and constraints between these variables formulated by relations over the appropriate domains; the  ...  We will survey results on the complexity of these computational tasks as a function of properties of the allowed constraint relations.  ...  sets of Boolean functions.  ... 
doi:10.1007/978-3-540-73001-9_80 fatcat:6plipcv4azccfa7b4zvvbpjake

meSAT: multiple encodings of CSP to SAT

Mirko Stojadinović, Filip Marić
2014 Constraints  
One approach for solving Constraint Satisfaction Problems (CSP) (and related Constraint Optimization Problems (COP)) involving integer and Boolean variables is reduction to propositional satisfiability  ...  We also present a methodology for selecting a suitable encoding based on simple syntactic features of the input CSP instance.  ...  Acknowledgements The authors wish to thank Mladen Nikolić, Predrag Janičić, and anonymous reviewers for valuable comments on the first versions of this manuscript.  ... 
doi:10.1007/s10601-014-9165-7 fatcat:nrtlxcewrjdpdco7wdpiytabne

Optimal polynomial-time compression for Boolean Max CSP [article]

Bart M.P. Jansen, Michał Włodarczyk
2020 arXiv   pre-print
In the Boolean maximum constraint satisfaction problem - Max CSP(Γ) - one is given a collection of weighted applications of constraints from a finite constraint language Γ, over a common set of variables  ...  , and the goal is to assign Boolean values to the variables so that the total weight of satisfied constraints is maximized.  ...  Our approach does not transfer even to the case with a domain of size 3, since there is no unique way to represent functions {0, 1, 2} k → {0, 1} as polynomials.  ... 
arXiv:2002.03443v1 fatcat:4yag7aw6tncytk4sa3jpbak7de
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