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

Tobias Friedrich, Ralf Rothenberger
2019 Proceedings of the Twenty-Eighth International Joint Conference on Artificial Intelligence  
This result generalizes Friedgut's 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.  ... 
doi:10.24963/ijcai.2019/853 dblp:conf/ijcai/0001R19 fatcat:x6mnkhzes5br3pc4kfq2oq6wiy

The Satisfiability Threshold for Non-Uniform Random 2-SAT

Tobias Friedrich, Ralf Rothenberger, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming  
We study 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.  ... 
doi:10.4230/lipics.icalp.2019.61 dblp:conf/icalp/0001R19 fatcat:dpwdbhpo75gdxmm2tea5fialua

The Satisfiability Threshold for Non-Uniform Random 2-SAT [article]

Tobias Friedrich, Ralf Rothenberger
2019 arXiv   pre-print
We study 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.  ... 
arXiv:1904.02027v1 fatcat:65jlbx6xobfkfjms76o4muzume

Phase Transitions for Scale-Free SAT Formulas

Tobias Friedrich, Anton Krohmer, Ralf Rothenberger, Andrew Sutton
2017 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
Recently, a number 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.  ... 
doi:10.1609/aaai.v31i1.11133 fatcat:64b4xw3zuba3re3oqtxuhuxtem

Upper Bound on the Satisfiability Threshold of Regular Random (k,s)-SAT Problem

Jincheng Zhou, Daoyun Xu
2016 International Journal of Innovative Computing, Information and Control  
Furthermore, it is also below 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  ... 
doi:10.24507/ijicic.12.02.477 fatcat:cjpi7ckokbdarkgqrwp7gq7rum

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

Phase Transitions and all that [article]

Gabriel Istrate
2005 arXiv   pre-print
It has been subsequently split into two papers, 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  ... 
arXiv:cs/0211012v2 fatcat:ojkdnqvi4jhrzlqclksqqnnvym

A sharp threshold for the renameable-Horn and the q-Horn properties

Nadia Creignou, Hervé Daudé, John Franco
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.  ... 
doi:10.1016/j.dam.2005.05.005 fatcat:6ovzrhs5zbestiz7fvvqdoqvvq

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

Gabriel Istrate, Stefan Boettcher, Allon G. Percus
2005 Annals of Mathematics and Artificial Intelligence  
We present several further results that add weight to 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.  ... 
doi:10.1007/s10472-005-7033-2 fatcat:u43fbvbrajczvmn5wmb4tkwwsm

Spines of Random Constraint Satisfaction Problems: Definition and Connection with Computational Complexity [article]

Gabriel Istrate, Stefan Boettcher, Allon G. Percus
2005 arXiv   pre-print
We present several further results that add weight to 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.  ... 
arXiv:cs/0503082v1 fatcat:wfqggbv43napzheyyiiwwrdmnu

Bounds on the Satisfiability Threshold for Power Law Distributed Random SAT [article]

Tobias Friedrich, Anton Krohmer, Ralf Rothenberger, Thomas Sauerwald, Andrew M. Sutton
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=(2k-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.  ... 
arXiv:1706.08431v1 fatcat:qkylsg27jfbdhe3u3kefylswda

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

Will Perkins
2013 arXiv   pre-print
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.  ...  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.  ... 
arXiv:1209.5313v2 fatcat:a3v2gx46rjgdhh4oc7mlggqxp4

Arithmetic Progression Hypergraphs: Examining the Second Moment Method [article]

Michael Mitzenmacher
2017 arXiv   pre-print
Our main result is to show that second moment arguments 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.  ... 
arXiv:1712.01781v1 fatcat:yhzuek7nvfes7crvvdsels5chy

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).  ...  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.  ... 
doi:10.1007/s10817-005-9012-z fatcat:ba6yzzc655fu5dwaqtjyfrniqu

Random k-SAT: Two Moments Suffice to Cross a Sharp Threshold [article]

Dimitris Achlioptas, Cristopher Moore
2003 arXiv   pre-print
As a corollary, we establish that 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.  ... 
arXiv:cond-mat/0310227v1 fatcat:5trrtdfkfjhajkecg5clqxaedi
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