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

Olivier Dubois, Yacine Boufkhad, Jacques Mandler
2002 arXiv   pre-print
We present a new structural (or syntatic) approach for estimating 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  ... 
arXiv:cs/0211036v1 fatcat:o522jarxcnhvjcgnvcfs33qnny

Phase Transitions for Scale-Free SAT Formulas

Tobias Friedrich, Anton Krohmer, Ralf Rothenberger, Andrew Sutton
Similar to 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.  ... 
doi:10.1609/aaai.v31i1.11133 fatcat:64b4xw3zuba3re3oqtxuhuxtem

Why almost all satisfiable $k$-CNF formulas are easy

Amin Coja-Oghlan, Michael Krivelevich, Dan Vilenchik
2007 Discrete Mathematics & Theoretical Computer Science  
We rigorously analyze 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.  ... 
doi:10.46298/dmtcs.3538 fatcat:gx5iozuryffi7nfoqc7mxrzxiy

Solution space heterogeneity of the randomK-satisfiability problem: Theory and simulations

Haijun Zhou
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).  ... 
doi:10.1088/1742-6596/233/1/012011 fatcat:k2j54j47afgajcfscj3euzpee4

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

Will Perkins
2013 arXiv   pre-print
We study an Achlioptas-process version of 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  ... 
arXiv:1209.5313v2 fatcat:a3v2gx46rjgdhh4oc7mlggqxp4

Glassy behavior and jamming of a random walk process for sequentially satisfying a constraint satisfaction formula

Haijun Zhou
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] .  ... 
doi:10.1140/epjb/e2010-00021-x fatcat:ajfvop7pfjfvjc5gjzadlpwlm4

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

Danila A. Sinopalnikov
2005 Lecture Notes in Computer Science  
We also prove an upper bound on 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.  ... 
doi:10.1007/11527695_21 fatcat:tpxrhql54bhsdi4agldswhdxoy

Regular Random k-SAT: Properties of Balanced Formulas

Yacine Boufkhad, Olivier Dubois, Yannet Interian, Bart Selman
2006 Journal of automated reasoning  
We consider a model 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).  ...  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 ) = 3αn/2 ! ( 3α/2 + 1)! µn 3α/2 ! (1−µ)n 2 ( 3α/2 + 1)! 2µn 3α/2 ! 2(1−µ)n (3αn)!  ... 
doi:10.1007/s10817-005-9012-z fatcat:ba6yzzc655fu5dwaqtjyfrniqu

Hiding Solutions in Random Satisfiability Problems: A Statistical Mechanics Approach

W. Barthel, A. K. Hartmann, M. Leone, F. Ricci-Tersenghi, M. Weigt, R. Zecchina
2002 Physical Review Letters  
On 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 .  ... 
doi:10.1103/physrevlett.88.188701 pmid:12005728 fatcat:lbqrp62sgjhlrjf5bhnamgzgyi

Communities of solutions in single solution clusters of a randomK-satisfiability formula

Haijun Zhou, Hui Ma
2009 Physical Review E  
This paper explores single solution clusters of 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.  ... 
doi:10.1103/physreve.80.066108 pmid:20365232 fatcat:dh5ui33oozahlgw7hxdmtqj62m

A review of the Statistical Mechanics approach to Random Optimization Problems [article]

Fabrizio Altarelli, Remi Monasson, Guilhem Semerjian, Francesco Zamponi
2008 arXiv   pre-print
We review 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] .  ... 
arXiv:0802.1829v1 fatcat:wv65adl4v5cmzjxon6unr7mg6q

Faster Random k-CNF Satisfiability

Andrea Lincoln, Adam Yedidia, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming  
We describe an algorithm to solve 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.  ... 
doi:10.4230/lipics.icalp.2020.78 dblp:conf/icalp/LincolnY20 fatcat:v6lxpmxk2nge3n6qt6rpuos72m

On the Empirical Time Complexity of Scale-Free 3-SAT at the Phase Transition [chapter]

Thomas Bläsius, Tobias Friedrich, Andrew M. Sutton
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  ... 
doi:10.1007/978-3-030-17462-0_7 fatcat:tgbcl5pbjffipgrgxvu7xhjjvy

Critical behavior in the computational cost of satisfiability testing

Bart Selman, Scott Kirkpatrick
1996 Artificial Intelligence  
There are intriguing similarities 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.  ... 
doi:10.1016/0004-3702(95)00056-9 fatcat:nwmqyvj7wbbwxedbbzfarlpj2q

Reweighted Belief Propagation and Quiet Planting for Random K-SAT

Florent Krzakala, Marc Mézard, Lenka Zdeborová
2014 Journal on Satisfiability, Boolean Modeling and Computation  
We are hence able to generate at 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  ... 
doi:10.3233/sat190096 fatcat:locashkltngn5cixxepvnxh6va
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