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Typical random 3-SAT formulae and the satisfiability threshold
[article]

2002
*
arXiv
*
pre-print

We present a new structural (or syntatic) approach for estimating

arXiv:cs/0211036v1
fatcat:o522jarxcnhvjcgnvcfs33qnny
*the**satisfiability**threshold*of*random**3*-*SAT**formulae*. ...*The*method combines well with other techniques,*and*also applies to other problems, such as*the**3*-colourability of*random*graphs. ... Experiments strongly suggest that*satisfiability*of*random**3*-*SAT**formulae*(*the**3*-*SAT*problem) exhibits a sharp*threshold*or a phase transition as a function of a control parameter,*the*ratio c of*the*number ...##
###
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
*

Similar to

doi:10.1609/aaai.v31i1.11133
fatcat:64b4xw3zuba3re3oqtxuhuxtem
*the*uniform model, on*formulas*with k>=*3*, we find that*the*phase transition regime corresponds to a set of*formulas*that are difficult to solve by backtracking search. ... Recently, a number of non-uniform*random**satisfiability*models have been proposed that are closer to practical*satisfiability*problems in some characteristics. ...*The*study of*random**SAT**formulas*is concerned with characterizing probability distributions over*satisfiability**formulas*. ...##
###
Why almost all satisfiable $k$-CNF formulas are easy

2007
*
Discrete Mathematics & Theoretical Computer Science
*

We rigorously analyze

doi:10.46298/dmtcs.3538
fatcat:gx5iozuryffi7nfoqc7mxrzxiy
*the*structure of*the*space of*satisfying*assignments of a*random**formula*in that distribution, showing that basically all*satisfying*assignments are clustered in one cluster,*and*... International audience Finding a*satisfying*assignment for a $k$-CNF*formula*$(k \geq*3*)$, assuming such exists, is a notoriously hard problem. ... As we already mentioned,*the*event of a*random**formula*in P n,m being*satisfiable*, when m/n is some constant above*the**satisfiability**threshold*, is very unlikely. ...##
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Solution space heterogeneity of the randomK-satisfiability problem: Theory and simulations

2010
*
Journal of Physics, Conference Series
*

*The*

*random*K-

*satisfiability*(K-

*SAT*) problem is an important problem for studying

*typical*-case complexity of NP-complete combinatorial satisfaction; it is also a representative model of finite-connectivity ... In this paper we review our recent efforts on

*the*solution space fine structures of

*the*

*random*K-

*SAT*problem. ... This work was partially supported by

*the*National Science Foundation of China (Grant numbers 10774150

*and*10834014)

*and*

*the*China 973-Program (Grant number 2007CB935903). ...

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

2013
*
arXiv
*
pre-print

We study an Achlioptas-process version of

arXiv:1209.5313v2
fatcat:a3v2gx46rjgdhh4oc7mlggqxp4
*the**random*k-*SAT*process: a bounded number of k-clauses are drawn uniformly at*random*at each step,*and*exactly one added to*the*growing*formula*according to a ... In particular, while a rule to delay*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*. ... A second central open question in this area is whether or not*random*k-*SAT**formulae*(k ≥*3*) at or near*the**satisfiability**threshold*are computationally hard (2-*SAT*is solvable in polynomial time in*the*...##
###
Glassy behavior and jamming of a random walk process for sequentially satisfying a constraint satisfaction formula

2010
*
European Physical Journal B : Condensed Matter Physics
*

*Random*K-

*satisfiability*(K-

*SAT*) is a model system for studying

*typical*-case complexity of combinatorial optimization. ... For

*random*

*3*-

*SAT*,

*the*jamming transition point α_j is larger than

*the*solution space clustering transition point α_d,

*and*its value can be predicted by a long-range frustration mean-field theory. ... As a comparison,

*the*

*satisfiability*

*threshold*of

*the*

*random*

*3*-

*SAT*problem is α s ≈ 4.27 [7, 19] ,

*and*that of

*the*

*random*4-

*SAT*problem is α s ≈ 9.93 [19] . ...

##
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Satisfiability Threshold of the Skewed Random k-SAT
[chapter]

2005
*
Lecture Notes in Computer Science
*

We also prove an upper bound on

doi:10.1007/11527695_21
fatcat:tpxrhql54bhsdi4agldswhdxoy
*the**satisfiability**threshold**and*a linear lower bound on*the*number of literals in*satisfying*partial assignments of skewed*random*k-*SAT**formulas*. ...*The*essential feature of*the**random*k-*SAT*is that positive*and*negative literals occur with equal probability in a*random**formula*. ... We proved an upper bound on*the**satisfiability**threshold**and*a lower bound on*the*number of literals in*satisfying*partial assignments for a skewed*random*k-*SAT**formula*. ...##
###
Regular Random k-SAT: Properties of Balanced Formulas

2006
*
Journal of automated reasoning
*

We consider a model for generating

doi:10.1007/s10817-005-9012-z
fatcat:ba6yzzc655fu5dwaqtjyfrniqu
*random*k-*SAT**formulas*, in which each literal occurs approximately*the*same number of times in*the**formula*clauses (regular*random*k-*SAT*). ... We will show that for α > 3.78 w.h.p. 1*random*Regular*3*-*SAT**formulas*are unsatisfiable. ... To sum up*the*probability that T*satisfies*a*random*Reg-*3*-*SAT**formula*is : Pr(T*satisfies*F ) =*3*αn/2 ! (*3*α/2 + 1)! µn*3*α/2 ! (1−µ)n 2 (*3*α/2 + 1)! 2µn*3*α/2 ! 2(1−µ)n (*3*αn)! ...##
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Hiding Solutions in Random Satisfiability Problems: A Statistical Mechanics Approach

2002
*
Physical Review Letters
*

On

doi:10.1103/physrevlett.88.188701
pmid:12005728
fatcat:lbqrp62sgjhlrjf5bhnamgzgyi
*the*basis of statistical mechanics results, we propose*random*generators of hard*and**satisfiable*instances for*the**3*-*satisfiability*problem (3SAT). ...*The*design of*the*hardest problem instances is based on*the*existence of a first order ferromagnetic phase transition*and**the*glassy nature of excited states. ...*The*transition point moves towards*the**random**3*-*SAT**threshold*a c ,*and**the*computational complexity increases with p 0 . ...##
###
Communities of solutions in single solution clusters of a randomK-satisfiability formula

2009
*
Physical Review E
*

This paper explores single solution clusters of

doi:10.1103/physreve.80.066108
pmid:20365232
fatcat:dh5ui33oozahlgw7hxdmtqj62m
*random**3*-*and*4-*SAT**formulas*through unbiased*and*biased*random*walk processes*and**the*replica-symmetric cavity method of statistical physics. ...*The*solution space of a K-*satisfiability*(K-*SAT*)*formula*is a collection of solution clusters, each of which contains all*the*solutions that are mutually reachable through a sequence of single-spin flips ... Results for*random**3*-*SAT*.-As an example, Fig. 1 (a) shows*the*simulation results for a*random**3*-*SAT**formula*of N = 10 6*and*α = 4.25. ...##
###
A review of the Statistical Mechanics approach to Random Optimization Problems
[article]

2008
*
arXiv
*
pre-print

We review

arXiv:0802.1829v1
fatcat:wv65adl4v5cmzjxon6unr7mg6q
*the*connection between statistical mechanics*and**the*analysis of*random*optimization problems, with particular emphasis on*the**random*k-*SAT*problem. ... We discuss*and*characterize*the*different phase transitions that are met in these problems, starting from basic concepts. ...*random**3*-*SAT*[55, 56]*and*XORSAT [33] . ...##
###
Faster Random k-CNF Satisfiability

2020
*
International Colloquium on Automata, Languages and Programming
*

We describe an algorithm to solve

doi:10.4230/lipics.icalp.2020.78
dblp:conf/icalp/LincolnY20
fatcat:v6lxpmxk2nge3n6qt6rpuos72m
*the*problem of Boolean CNF-*Satisfiability*when*the*input*formula*is chosen randomly. We build upon*the*algorithms of Schöning 1999*and*Dantsin et al. in 2002. ...*The*Schöning algorithm works by trying many possible*random*assignments,*and*for each one searching systematically in*the*neighborhood of that assignment for a*satisfying*solution. ... Preliminaries In this section we will give*the*definition of*random*k-CNF*Satisfiability*(*random*k-CNF*SAT*) at*the**threshold*. ...##
###
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*...

##
###
Critical behavior in the computational cost of satisfiability testing

1996
*
Artificial Intelligence
*

There are intriguing similarities

doi:10.1016/0004-3702(95)00056-9
fatcat:nwmqyvj7wbbwxedbbzfarlpj2q
*and*differences between these effects*and*kinetic phenomena studied in statistical physics, in glasses*and*in spin glasses. Fig. 1. (a)*Threshold*for*random**3*-*SAT*. ... In previous work, we employed finite-size scaling, a method from statistical mechanics, to explore*the*crossover from*the**SAT*regime of k-*SAT*, where almost all randomly generated expressions are*satisfiable*... We also thank Jimi Crawford, Ian Gent, Tad Hogg, Robert Schrag, Toby Walsh,*and*Colin Williams for useful comments*and*discussions. ...##
###
Reweighted Belief Propagation and Quiet Planting for Random K-SAT

2014
*
Journal on Satisfiability, Boolean Modeling and Computation
*

We are hence able to generate at

doi:10.3233/sat190096
fatcat:locashkltngn5cixxepvnxh6va
*the*same time a*typical**random**SAT*instance*and*one of its solutions. ... region of parameters in*the**random*K-*satisfiability*problem. ... Most importantly we give further theoretical justification for why*the*λ * -planted*random**3*-*SAT**formulas*are (together with benchmarks from [16] )*the*hardest*satisfiable**formulas*on*the*market of hard ...
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