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Sharpness of the Satisfiability Threshold for Non-Uniform Random k-SAT

2019
*
Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence
*

This result generalizes Friedgut's

doi:10.24963/ijcai.2019/853
dblp:conf/ijcai/0001R19
fatcat:x6mnkhzes5br3pc4kfq2oq6wiy
*sharpness*result from*uniform*to*non*-*uniform**random**k*-*SAT*and implies*sharpness**for**thresholds**of*a wide range*of**random**k*-*SAT*models with heterogeneous probability ... ., p_n) on*the*variable occurrences. Therefore, we call it*non*-*uniform**random**k*-*SAT*. ...*The*goal*of*this paper is to show*the*second ingredient*for*proving*the**satisfiability**threshold*conjecture*for**non*-*uniform**random**k*-*SAT*,*sharpness**of**the**satisfiability**threshold*. ...##
###
The Satisfiability Threshold for Non-Uniform Random 2-SAT

2019
*
International Colloquium on Automata, Languages and Programming
*

We study

doi:10.4230/lipics.icalp.2019.61
dblp:conf/icalp/0001R19
fatcat:dpwdbhpo75gdxmm2tea5fialua
*satisfiability**thresholds**of**non*-*uniform**random*2-*SAT*with n variables and m clauses and with an arbitrary probability distribution (pi) i∈ [n] with p1 p2 . . . pn > 0 over*the*n variables. ... We show*for*p and that it is coarse.*For*p 2 1 = o n i=1 p 2 i we show that there is a*sharp**satisfiability**threshold*at m = n i=1 p 2 i −1 . ... Note that these three cases give us a complete dichotomy*of*coarseness and*sharpness**for**the**satisfiability**threshold**of**non*-*uniform**random*2-*SAT*. ...##
###
The Satisfiability Threshold for Non-Uniform Random 2-SAT
[article]

2019
*
arXiv
*
pre-print

We study

arXiv:1904.02027v1
fatcat:65jlbx6xobfkfjms76o4muzume
*satisfiability**thresholds**of**non*-*uniform**random*2-*SAT*with n variables and m clauses and with an arbitrary probability distribution (p_i)_i∈[n] with p_1 > p_2 >...> p_n > 0 over*the*n variables ...*For*p_1^2=o(∑_i=1^n p_i^2) we show that there is a*sharp**satisfiability**threshold*at m=(∑_i=1^n p_i^2)^-1. ... Note that these three cases give us a complete dichotomy*of*coarseness and*sharpness**for**the**satisfiability**threshold**of**non*-*uniform**random*2-*SAT*. ...##
###
Phase Transitions for Scale-Free SAT Formulas

2017
*
PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE
*

Recently, a number

doi:10.1609/aaai.v31i1.11133
fatcat:64b4xw3zuba3re3oqtxuhuxtem
*of**non*-*uniform**random**satisfiability*models have been proposed that are closer to practical*satisfiability*problems in some characteristics. ...*For*scale-free formulas with clauses*of*length*k*=2, we give a lower bound on*the*phase transition*threshold*as a function*of**the*scaling parameter. ... We sketch a proof*for*a lower bound on*the*location*of**the**threshold**for**the*case*of**k*= 2. ...##
###
Upper Bound on the Satisfiability Threshold of Regular Random (k,s)-SAT Problem

2016
*
International Journal of Innovative Computing, Information and Control
*

Furthermore, it is also below

doi:10.24507/ijicic.12.02.477
fatcat:cjpi7ckokbdarkgqrwp7gq7rum
*the*asymptotic bound*of**the**uniform**k*-*SAT*problem, which is known as 2*k*log 2 − (log 2 + 1)/2 + o*k*(1)*for*large*k*. ... Experiment results also indicate that*the**threshold**for*strictly regular*random*(*k*, s)-*SAT*problem is very close to our theoretical upper bound, and*the*regular*random*(*k*, s)-*SAT*instances generated by ...*The*authors would like to thank*the*anonymous referees*for*their valuable suggestions and comments. This ...##
###
On the Empirical Time Complexity of Scale-Free 3-SAT at the Phase Transition
[chapter]

2019
*
Lecture Notes in Computer Science
*

*The*hardness

*of*formulas at

*the*solubility phase transition

*of*

*random*propositional

*satisfiability*(

*SAT*) has been intensely studied

*for*decades both empirically and theoretically. ...

*The*main contribution

*of*this paper utilizes

*the*scale-free

*random*3-

*SAT*model to isolate heterogeneity as an important factor in

*the*scaling discrepancy between complete and SLS solvers at

*the*

*uniform*... are typically recognized

*for*their performance on

*uniform*

*random*

*k*-

*SAT*, do not benefit much (if at all) from heterogeneity. (3)

*Random*instances (even heterogeneous ones) close to

*the*

*satisfiability*

*threshold*...

##
###
Phase Transitions and all that
[article]

2005
*
arXiv
*
pre-print

It has been subsequently split into two papers,

arXiv:cs/0211012v2
fatcat:ojkdnqvi4jhrzlqclksqqnnvym
*the*corrected (and accepted*for*publication) versions appear in*the*archive as papers cs.CC/0503082 and cs.DM/0503083. ...*The*paper (as posted originally) contains several errors. ...*The*proof*of*Claim 6 (and*of*item 2.*of*Theorem 5) follows: since*for*any clause*K**of*one*of**the*original constraints µ(*K*) = 1, since µ( ) > η 1 · n and since w.l.o.g. 0 < d < η 1 (otherwise replace d ...##
###
A sharp threshold for the renameable-Horn and the q-Horn properties

2005
*
Discrete Applied Mathematics
*

*The*

*sharp*

*Satisfiability*

*threshold*is well known

*for*

*random*

*k*-

*SAT*formulas and is due to certain minimality and monotonic properties mentioned in this manuscript and reported in Chandru and Hooker [J. ... Whereas

*the*

*Satisfiability*

*threshold*is on

*the*probability that a

*satisfying*assignment exists, we find that

*sharp*

*thresholds*also may be determined

*for*certain formula structures,

*for*example,

*the*probability ... Friedgut [20] , with an appendix by Bourgain, proved that

*k*-

*SAT*exhibits a

*sharp*

*threshold*

*for*

*k*2 but without specifying its location. ...

##
###
Spines of random constraint satisfaction problems: definition and connection with computational complexity

2005
*
Annals of Mathematics and Artificial Intelligence
*

We present several further results that add weight to

doi:10.1007/s10472-005-7033-2
fatcat:u43fbvbrajczvmn5wmb4tkwwsm
*the*intuition that*random*constraint satisfaction problems with a*sharp**threshold*and a continuous spine are "qualitatively similar to*random*2-*SAT*...*The*two phenomena have a common underlying cause:*the*emergence*of*"large" (linear size) minimally unsatisfiable subformulas*of*a*random*formula at*the**satisfiability*phase transition. ... One can give an explicitly defined class*of**satisfiability*problems*for*which*the*previous result applies: Theorem 3 Let*k*≥ 2 and let C be such that*SAT*(C) has a*sharp**threshold*. ...##
###
Spines of Random Constraint Satisfaction Problems: Definition and Connection with Computational Complexity
[article]

2005
*
arXiv
*
pre-print

We present several further results that add weight to

arXiv:cs/0503082v1
fatcat:wfqggbv43napzheyyiiwwrdmnu
*the*intuition that*random*constraint satisfaction problems with a*sharp**threshold*and a continuous spine are "qualitatively similar to*random*2-*SAT*...*The*two phenomena have a common underlying cause:*the*emergence*of*"large" (linear size) minimally unsatisfiable subformulas*of*a*random*formula at*the**satisfiability*phase transition. ... One can give an explicitly defined class*of**satisfiability*problems*for*which*the*previous result applies: Theorem 3 Let*k*≥ 2 and let C be such that*SAT*(C) has a*sharp**threshold*. ...##
###
Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT
[article]

2017
*
arXiv
*
pre-print

*The*average-case analysis

*of*

*SAT*has triggered

*the*development

*of*sophisticated rigorous and

*non*-rigorous techniques

*for*analyzing

*random*structures. ... We study

*random*

*k*-

*SAT*on n variables, m=Θ(n) clauses, and a power law distribution on

*the*variable occurrences with exponent β. We observe a

*satisfiability*

*threshold*at β=(2

*k*-1)/(

*k*-1). ...

*For*

*the*concrete power law distribution in Definition 2.3,

*the*true

*threshold*is much smaller. In fact, it appears to be below

*the*

*satisfiability*

*threshold*

*for*

*uniform*

*random*

*SAT*. ...

##
###
Random k-SAT and the Power of Two Choices
[article]

2013
*
arXiv
*
pre-print

In particular, while a rule to delay

arXiv:1209.5313v2
fatcat:a3v2gx46rjgdhh4oc7mlggqxp4
*the*2-*SAT**threshold*was known previously, this is*the*first proof*of*a rule to shift*the**threshold**of**k*-*SAT**for**k*>= 3. ... We prove*the*existence*of*a rule that shifts*the**satisfiability**threshold*. This extends a well-studied area*of*probabilistic combinatorics (Achlioptas processes) to*random*CSP's. ... We have discussed bounds on*the**satisfiability**threshold**of*Achlioptas processes here and mentioned Friedgut's result on*the**sharpness**of**the**k*-*SAT**threshold*. ...##
###
Arithmetic Progression Hypergraphs: Examining the Second Moment Method
[article]

2017
*
arXiv
*
pre-print

Our main result is to show that second moment arguments

arXiv:1712.01781v1
fatcat:yhzuek7nvfes7crvvdsels5chy
*for*3-NAE-*SAT*and 2-coloring*of*3-regular hypergraphs extend to*the*double hashing setting. ... We leave several open problems related to these quasi-*random*hypergraphs and*the**thresholds**of*associated problem variations. ... Acknowledgments*The*author thanks Cris Moore*for*several helpful discussions, and Yufei Zhao*for*explaining results regarding monochromatic colorings*of*arithmetic progressions*of*length 3. ...##
###
Regular Random k-SAT: Properties of Balanced Formulas

2006
*
Journal of automated reasoning
*

We consider a model

doi:10.1007/s10817-005-9012-z
fatcat:ba6yzzc655fu5dwaqtjyfrniqu
*for*generating*random**k*-*SAT*formulas, in which each literal occurs approximately*the*same number*of*times in*the*formula clauses (regular*random**k*-*SAT*). ... Our experimental results show that such regular*random**k*-*SAT*instances are much harder than*the*usual*uniform**random**k*-*SAT*problems. ...*The**threshold**for*regular 2-*SAT*is at*the*same ratio*of*α = 1 as*for**the**uniform**random*2-*SAT*model. So, only*for**k*> 2, do*the*properties*of**the*models diverge in an interesting way. ...##
###
Random k-SAT: Two Moments Suffice to Cross a Sharp Threshold
[article]

2003
*
arXiv
*
pre-print

As a corollary, we establish that

arXiv:cond-mat/0310227v1
fatcat:5trrtdfkfjhajkecg5clqxaedi
*the**threshold**for**random**k*-*SAT*is*of*order Theta(2^*k*), resolving a long-standing open problem. ... Specifically, we prove that*the**threshold**for*both*random*hypergraph 2-colorability (Property B) and*random*Not-All-Equal*k*-*SAT*is 2^*k*-1 ln 2 -O(1). ... , and to Remi Monasson*for*discussions on*the*replica method. ...
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