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A remark on random 2-SAT

Andreas Goerdt
1999 Discrete Applied Mathematics  
It is known that almost all 2-CNF-formulas where the number of clauses is n(1 − ); ¿ 0 a constant, n the number of variables are satisÿable. For n(1 + ) clauses they are known to be unsatisÿable.  ...  We strengthen the ÿrst result and show that for each = (n)¿(1=(ln n)) 1=2 (hence may approach 0) the ÿrst of the results above still holds. ? (A. Goerdt) 0166-218X/99/$ -see front matter ?  ...  A formula is unsatisÿable i its formula graph has a cycle containing 2 literals x und x . We assume that we have an ordering {x 1 ; x 1 } ¡ {x 2 ; x 2 } : : : on our literals.  ... 
doi:10.1016/s0166-218x(99)00034-7 fatcat:cnlijvzmkrgglofry57exbi5he

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  ...  agree on all but a small, though linear, number of variables.  ...  For lack of space most propositions are given without a proof. 2 Properties of a Random Instance from P sat n,m In this section we analyze the structure of a typical formula in P sat n,m .  ... 
doi:10.46298/dmtcs.3538 fatcat:gx5iozuryffi7nfoqc7mxrzxiy

When does the giant component bring unsatisfiability?

Michael Molloy
2008 Combinatorica  
We study random constraint satisfaction problems using the wide class of models introduced by the author [36], which includes various forms of random SAT and other wellstudied problems.  ...  We determine precisely which of these models remain almost surely satisfiable when the number of clauses is increased beyond the point at which a giant component appears in the underlying constraint hypergraph  ...  [1, 2, 8, 10, 15, 23, 29, 31, 33] ) random NAE-SAT [3] and the colourability of random graphs [4, 6, 7, 9, 18, 22, 32] .  ... 
doi:10.1007/s00493-008-2123-5 fatcat:2g6f52poibhbpptfvg6nafkgna

On Average Case Complexity of SAT for Symmetric Distribution

JOHANN A. MAKOWSKY, ABRAHAM SHARELL
1995 Journal of Logic and Computation  
A randomized decision problem is a tuple hD; i (formally hD; S; i) with  ...  DistNP complete) distributions for SAT, if one considers only deterministic reductions.  ...  Based on recent results about the satis ability threshold of 2-SAT we remark (52), that for random 3-CNF instances with more than n 2 clauses short resolution proofs exist with high probability.  ... 
doi:10.1093/logcom/5.1.71 fatcat:jx5qrwr7f5bv7kxey4zxwpag6m

Local search for Boolean Satisfiability with configuration checking and subscore

Shaowei Cai, Kaile Su
2013 Artificial Intelligence  
The CC strategy results in a simple local search algorithm for SAT called Swcc, which shows promising experimental results on random 3-SAT instances, and outperforms TNM, the winner of SAT Competition  ...  Although the SAT community has made great achievements in solving random 3-SAT instances, the progress lags far behind on random k-SAT instances with k > 3.  ...  We would like to thank the anonymous reviewers for their helpful comments on earlier versions of this paper.  ... 
doi:10.1016/j.artint.2013.09.001 fatcat:ul7fjqbg6bcetcom5rmo6p266e

On the Satisfiability Threshold of Random Community-Structured SAT

Dina Barak-Pelleg, Daniel Berend
2018 Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence  
One type of instances arises when the clauses are chosen uniformly randomly \textendash{} random SAT.  ...  There is as yet no generally accepted model of industrial SAT, and we confine ourselves to one of the more common features of industrial SAT: the set of variables consists of a number of disjoint communities  ...  Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence  ... 
doi:10.24963/ijcai.2018/174 dblp:conf/ijcai/Barak-PellegB18 fatcat:6x2f2jkzpjdk5mptil5kcr5gri

On quantum versions of record-breaking algorithms for SAT

Evgeny Dantsin, Vladik Kreinovich, Alexander Wolpert
2005 ACM SIGACT News  
It is well known that a straightforward application of Grover's quantum search algorithm enables to solve SAT in O(2 n/2 ) steps.  ...  Ambainis (SIGACT News, 2004) observed that it is possible to use Grover's technique to similarly speed up a sophisticated algorithm for solving 3-SAT.  ...  Can we actually use quantum computing to solve SAT faster than in time ∼ 2 n/2 ? In this note, we discuss some aspects of this question. Remark.  ... 
doi:10.1145/1107523.1107524 fatcat:tbn5ej6u5zbyjiwe6rhv6rqyuy

NP is as easy as detecting unique solutions

L.G. Valiant, V.V. Vazirani
1986 Theoretical Computer Science  
We show that the problems of distinguishing between instances of SAT having zero or one solution, or of finding solutions to instances of SAT having a unique solution, are as hard as SAT, under randomized  ...  For example, computing the parity of the number of solutions of a SAT formula is shown to be NP-hard, and deciding if a SAT formula has a unique solution is shown to be DP-hard, under randomized reduction  ...  On the other hand, a deterministic reduction from SAT to UNIQUE SAT will make UNIQUE SAT complete in D p.  ... 
doi:10.1016/0304-3975(86)90135-0 fatcat:msrd4dyx6fgrhfmx3dgstfj2z4

On the Scaling Window of Model RB [article]

Chunyan Zhao, Ke Xu, Zhiming Zheng
2008 arXiv   pre-print
This paper analyzes the scaling window of a random CSP model (i.e. model RB) for which we can identify the threshold points exactly, denoted by r_cr or p_cr.  ...  For this model, we establish the scaling window W(n,δ)=(r_-(n,δ), r_+(n,δ)) such that the probability of a random instance being satisfiable is greater than 1-δ for rr_+(n,δ).  ...  Motivated by the work on the scaling window of random 2-SAT, in this paper, we study the scaling window of model RB and obtain that W (n, δ) = (r cr −Θ( 1 n 1−ε ln n ), r cr +Θ( 1 n ln n )).  ... 
arXiv:0801.3871v1 fatcat:vmsaoerbzncqtfwvib74eugsia

Classical-quantum mixing in the random 2-satisfiability problem

Ionut-Dragos Potirniche, C. R. Laumann, S. L. Sondhi
2015 Physical Review A. Atomic, Molecular, and Optical Physics  
For the simplest 2-SAT/2-QSAT ensemble we find the exact boundary that separates SAT and UNSAT instances.  ...  We do so by establishing coincident lower and upper bounds, in the limit of large instances, on the extent of the UNSAT and SAT regions, respectively.  ...  Bardoscia for useful discussions and to A. Scardicchio and R. Moessner for collaboration on related work.  ... 
doi:10.1103/physreva.92.040301 fatcat:e7od6gm56jggnp4fyjhbxmtdjm

Simpler Partial Derandomization of PPSZ for k-SAT [article]

S. Cliff Liu
2020 arXiv   pre-print
We give a simpler derandomization of the best known k-SAT algorithm PPSZ [FOCS'97, JACM'05] for k-SAT with sub-exponential number of solutions.  ...  Our algorithm and theorem also have a nice byproduct: It outperforms the current fastest deterministic k-SAT algorithm when the formula has moderately exponential number of solutions.  ...  Therefore, by Theorem 2 and a routine argument, we obtain a randomized algorithm for k-SAT with one-sided error, whose upper bound of the running time is O(2 (1−λ)n ).  ... 
arXiv:2001.06536v1 fatcat:f6x2ca4aivdazjvur3ztx7pwdq

Results on a Super Strong Exponential Time Hypothesis

Nikhil Vyas, Ryan Williams
2020 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
It turns out that a well-known algorithm from the literature on SAT algorithms does the job: the PPZ algorithm of Paturi, Pudlak, and Zane (1998).2.  ...  We give a randomized algorithm which refutes the Super-Strong ETH for the case of random k-SAT and planted k-SAT for any clause-to-variable ratio.  ...  Given a random k-SAT instance F sampled from R + (n, k, m), we can find a solution in 2 n(1−Ω( log k k )) time whp. Remark 1.  ... 
doi:10.1609/aaai.v34i09.7125 fatcat:jocm2ycqdrcobed5yopysjdotm

Exhaustive enumeration unveils clustering and freezing in the random 3-satisfiability problem

John Ardelius, Lenka Zdeborová
2008 Physical Review E  
We study geometrical properties of the complete set of solutions of the random 3-satisfiability problem.  ...  We locate the freezing transition in the space of solutions which has been conjectured to be relevant in explaining the onset of computational hardness in random constraint satisfaction problems.  ...  Each clause contains a subset of K distinct variables chosen uniformly at random, and each clause forbids one random assignment of the K variables out of the 2 K possible ones.  ... 
doi:10.1103/physreve.78.040101 pmid:18999364 fatcat:d4m4emfu3fgetmxhsnke5ugq7a

2006 and 2007 Max-SAT Evaluations: Contributed Instances

Federico Heras, Javier Larrosa, Simon de Givry, Thomas Schiex, Ewald Speckenmeyer, Chu Min Li, Vasco Manquinho, Armando Tacchella
2008 Journal on Satisfiability, Boolean Modeling and Computation  
Finally, we present a quick reference table containing short descriptions of each set of problem instances.  ...  In this technical report we briefly describe the instances submitted to the 2006 and 2007 Max-SAT Evaluations. First, we introduce the instances that can be directly encoded as Max-SAT.  ...  However, some instances were submitted by other authors: • Random Max-2-SAT (U), random Max-3-SAT (U), random Weighted Max-2-SAT (W), random Weighted Max-3-SAT (W), random unweighted Partial Max-2-SAT  ... 
doi:10.3233/sat190046 fatcat:ilg56fel3fgyrmgciutoxaqf64

Random MAX SAT, random MAX CUT, and their phase transitions

Don Coppersmith, David Gamarnik, MohammadTaghi Hajiaghayi, Gregory B. Sorkin
2004 Random structures & algorithms (Print)  
For random graphs, a maximization version of the giant-component question behaves quite differently from 2-sat, but max cut behaves similarly.  ...  Specifically, random 2-sat formulas with clause/variable density less than 1 are almost always satisfiable, those with density greater than 1 are almost always unsatisfiable, and the "scaling window" is  ...  Summary of results We establish several properties of random max 2-sat, random max k-sat, and random max cut, focusing on 2-sat.  ... 
doi:10.1002/rsa.20015 fatcat:e56x63terffpxgjdy2wjciz3ri
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