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

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 ...

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

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 ...

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

2008
*
Lecture Notes in Computer Science
*

Pol(Γ) is

doi:10.1007/978-3-540-92800-3_2
fatcat:5ahjgjte5vfalii7u7rre6kdkq
*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. ...##
###
Counting Problems and Clones of Functions

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 ). ...

##
###
An approximation trichotomy for Boolean #CSP

2010
*
Journal of computer and system sciences (Print)
*

We give a trichotomy theorem for

doi:10.1016/j.jcss.2009.08.003
fatcat:sod7lmaym5eg7kz34bmg4hpk6e
*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*. ...##
###
Identifying Efficiently Solvable Cases of Max CSP
[chapter]

2004
*
Lecture Notes in Computer Science
*

Finally, we show that

doi:10.1007/978-3-540-24749-4_14
fatcat:k2fq6aljyzan3e2hn376u6zley
*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*...##
###
The Complexity of Problems for Quantified Constraints

2009
*
Theory of Computing Systems
*

Generalizing these results

doi:10.1007/s00224-009-9194-6
fatcat:if5enr6ltvhebeo4vimq32qzsi
*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. ...##
###
An approximation trichotomy for Boolean #CSP
[article]

2009
*
arXiv
*
pre-print

We give a trichotomy theorem for

arXiv:0710.4272v2
fatcat:dxdbyt357jg4refsyexdkzf3fi
*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*...##
###
Optimal Polynomial-Time Compression for Boolean Max CSP

2020
*
European Symposium on Algorithms
*

In

doi:10.4230/lipics.esa.2020.63
dblp:conf/esa/Jansen020
fatcat:vdmlgng22nebvltgqwmnfgmei4
*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. ...##
###
The complexity of weighted Boolean #CSP with mixed signs

2009
*
Theoretical Computer Science
*

Such a problem is parameterized by a set Γ

doi:10.1016/j.tcs.2009.06.003
fatcat:kghgqstmrff77blhqgijn6kkm4
*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}. ...##
###
The Complexity of Weighted Boolean #CSP with Mixed Signs
[article]

2009
*
arXiv
*
pre-print

Such a problem is parameterized by a set

arXiv:0812.4171v2
fatcat:muim3mn4w5dfvdoi5nhqldreeu
*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}. ...##
###
Tractability in constraint satisfaction problems: a survey

2015
*
Constraints
*

Acknowledgments We are grateful

doi:10.1007/s10601-015-9198-6
fatcat:fl7kxmceh5bqzpyd2c37rtohii
*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. ...##
###
Computational Complexity of Constraint Satisfaction
[chapter]

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*. ...

##
###
meSAT: multiple encodings of CSP to SAT

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. ...

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

2020
*
arXiv
*
pre-print

In

arXiv:2002.03443v1
fatcat:4yag7aw6tncytk4sa3jpbak7de
*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. ...
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