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Counting solutions to random CNF formulas [article]

Andreas Galanis and Leslie Ann Goldberg and Heng Guo and Kuan Yang
2021 arXiv   pre-print
We give the first efficient algorithm to approximately count the number of solutions in the random k-SAT model when the density of the formula scales exponentially with k.  ...  Instead, our algorithm harnesses a recent technique by Moitra to work for random formulas.  ...  As our goal is to count satisfying assignments of sparse random k-CNF formulas, where these degree bounds do not hold, but average degrees are small, it is natural to also choose Moitra's method in the  ... 
arXiv:1911.07020v4 fatcat:umza5sdnqrh5bafg4vbduxswvi

Counting Solutions to Random CNF Formulas

Andreas Galanis, Leslie Ann Goldberg, Heng Guo, Kuan Yang, Emanuela Merelli, Artur Czumaj, Anuj Dawar
2020 International Colloquium on Automata, Languages and Programming  
We give the first efficient algorithm to approximately count the number of solutions in the random k-SAT model when the density of the formula scales exponentially with k.  ...  Instead, our algorithm harnesses a recent technique by Moitra to work for random formulas with much higher densities.  ...  Then, for a particular marked variable 53:4 Counting Solutions to Random CNF Formulas v, we set up an LP.  ... 
doi:10.4230/lipics.icalp.2020.53 dblp:conf/icalp/GalanisG0Y20 fatcat:qwawdewkizb2jakztnzii5bixy

Counting Solutions to Random CNF Formulas

Andreas Galanis, Leslie Ann Goldberg, Heng Guo, Kuan Yang
2021 SIAM journal on computing (Print)  
We give the first efficient algorithm to approximately count the number of solutions in the random k-SAT model when the density of the formula scales exponentially with k.  ...  Instead, our algorithm harnesses a recent technique by Moitra to work for random formulas with much higher densities.  ...  As our goal is to count satisfying assignments of sparse random k-CNF formulas, where these degree bounds do not hold, but average degrees are small, it is natural to also choose Moitra's method in the  ... 
doi:10.1137/20m1351527 fatcat:zzdf76xssngipefh5khhqr73kq

The Hard Problems Are Almost Everywhere For Random CNF-XOR Formulas

Jeffrey M. Dudek, Kuldeep S. Meel, Moshe Y. Vardi
2017 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence  
On the theoretical front, we prove that the solution space of a random CNF-XOR formula 'shatters' at all nonzero XOR-clause densities into well-separated components, similar to the behavior seen in random  ...  CNF formulas known to be difficult for many SAT algorithms.  ...  We prove (in Section 4) that the solution space of random variable-width XOR formulas (and therefore of random k-CNF-XOR formulas) shatters.  ... 
doi:10.24963/ijcai.2017/84 dblp:conf/ijcai/DudekMV17 fatcat:l3d4acceencftpib24rheu7n34

On Hashing-Based Approaches to Approximate DNF-Counting [article]

Kuldeep S. Meel National University of Singapore
2017 arXiv   pre-print
When the constraints are expressed as DNF formulas, Monte Carlo-based techniques have been shown to provide a fully polynomial randomized approximation scheme (FPRAS).  ...  Given the success of hashing-based techniques for CNF constraints, it is natural to ask: Can hashing-based techniques provide an efficient FPRAS for DNF counting?  ...  Given a Boolean formula φ, the problem of propositional model counting , also referred to as #SAT, is to compute the number of solutions of φ [29] .  ... 
arXiv:1710.05247v1 fatcat:oy5tcwuqybd7xpeia6xmfaphge

Full CNF Encoding: The Counting Constraints Case

Olivier Bailleux, Yacine Boufkhad
2004 International Conference on Theory and Applications of Satisfiability Testing  
Many problems are naturally expressed using CNF clauses and boolean cardinality constraints.  ...  We show experimentally that our encoding allows zchaff to be competitive with pseudo-boolean solvers on some decision and optimization problems.  ...  Second, we present a DLL based optimization procedure that maximizes the number of variables fixed to 1 in a given subset of the variables belonging to a CNF formula, while satisfying the formula.  ... 
dblp:conf/sat/BailleuxB04 fatcat:i6my6motlvg25dr4cooc5j7d6q

Tinted, Detached, and Lazy CNF-XOR Solving and Its Applications to Counting and Sampling [chapter]

Mate Soos, Stephan Gocht, Kuldeep S. Meel
2020 Lecture Notes in Computer Science  
Given a Boolean formula, the problem of counting seeks to estimate the number of solutions of F while the problem of uniform sampling seeks to sample solutions uniformly at random.  ...  The past few years have witnessed the rise of hashing-based approaches that use XORbased hashing and employ SAT solvers to solve the resulting CNF formulas conjuncted with XOR constraints.  ...  The problem of propositional model counting is to compute |sol(F )| for a given CNF formula F .  ... 
doi:10.1007/978-3-030-53288-8_22 fatcat:5hkp4g2yx5d2pgpzekcym44nsi

An Approximation Algorithm for #k-SAT

Marc Thurley, Marc Herbstritt
2012 Symposium on Theoretical Aspects of Computer Science  
For 4-CNF input the bound increases to O(1.6155 n ). We further show how to obtain upper and lower bounds on the number of solutions to a CNF formula in a controllable way.  ...  For example, even stipulating sub-exponentially small error tolerance, the number of solutions to 3-CNF input formulas can be approximated in time O(1.5366 n ).  ...  Acknowledgments I would like to thank Martin Grohe for bringing this problem to my attention and for several helpful discussions on the topic.  ... 
doi:10.4230/lipics.stacs.2012.78 dblp:conf/stacs/Thurley12 fatcat:375aoaw36nclvckyjuip4bv4kq

Combining the k-CNF and XOR Phase-Transitions [article]

Jeffrey M. Dudek, Kuldeep S. Meel, Moshe Y. Vardi
2017 arXiv   pre-print
Recent universal hashing-based approaches to sampling and counting crucially depend on the runtime performance of SAT solvers on formulas expressed as the conjunction of both k-CNF and XOR constraints  ...  (known as k-CNF-XOR formulas), but the behavior of random k-CNF-XOR formulas is unexplored in prior work.  ...  Acknowledgments The authors would like to thank Dimitris Achlioptas for helpful discussions in the early stages of this project.  ... 
arXiv:1702.08392v1 fatcat:vyap6gbgnjedtm6t2vsbebljqi

An Approximation Algorithm for #k-SAT [article]

Marc Thurley
2011 arXiv   pre-print
For 4-CNF input the bound increases to O(1.6155^n). We further show how to obtain upper and lower bounds on the number of solutions to a CNF formula in a controllable way.  ...  For example, even stipulating sub-exponentially small error tolerance, the number of solutions to 3-CNF input formulas can be approximated in time O(1.5366^n).  ...  Preliminaries For a CNF formula F , let sat(F ) be the set of its solutions and #F = |sat(F )|. We shall always use n to denote the number of variables of a CNF formula under consideration.  ... 
arXiv:1107.2001v1 fatcat:4ti42kporvef7kgmm2qhlqotli

Computing the Density of States of Boolean Formulas [chapter]

Stefano Ermon, Carla P. Gomes, Bart Selman
2010 Lecture Notes in Computer Science  
In this paper we consider the problem of computing the density of states of a Boolean formula in CNF, a generalization of both MAX-SAT and model counting.  ...  We propose a novel Markov Chain Monte Carlo algorithm based on flat histogram methods that, despite the hardness of the problem, converges quickly to a very accurate solution.  ...  Random Formulas In this section we present a detailed study of the behavior of the DOS for random 3-SAT formulas as a function of the ratio clauses to variables α.  ... 
doi:10.1007/978-3-642-15396-9_6 fatcat:ba4b72rtbzayhg2bc3bfugse5i

Solving and Sampling with Many Solutions: Satisfiability and Other Hard Problems [article]

Jean Cardinal, Jerri Nummenpalo, Emo Welzl
2017 arXiv   pre-print
We investigate parameterizing hard combinatorial problems by the size of the solution set compared to all solution candidates.  ...  Our main result is a uniform sampling algorithm for satisfying assignments of 2-CNF formulas that runs in expected time O^*(ε^-0.617) where ε is the fraction of assignments that are satisfying.  ...  Acknowledgments We would like to thank Noga Alon and József Solymosi for discussions on the problem. We also thank the reviewers of IPEC 2017 for valuable remarks that improved the exposition.  ... 
arXiv:1708.01122v1 fatcat:dk4o2sovnvhb7anxj7iagu2era

Distribution-Aware Sampling and Weighted Model Counting for SAT

Supratik Chakraborty, Daniel Fremont, Kuldeep Meel, Sanjit Seshia, Moshe Vardi
2014 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
Given a CNF formula and a weight for each assignment of values tovariables, two natural problems are weighted model counting anddistribution-aware sampling of satisfying assignments.  ...  Due to the inherentcomplexity of the exact versions of the problems, interest has focusedon solving them approximately.  ...  Acknowledgments: The authors would like to thank Armando Solar-Lezama for generously providing a large set of benchmarks.  ... 
doi:10.1609/aaai.v28i1.8990 fatcat:474lujaf4zccrdy56rlclkskqi

Solving Stochastic Boolean Satisfiability under Random-Exist Quantification

Nian-Ze Lee, Yen-Shi Wang, Jie-Hong R. Jiang
2017 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence  
Experimental results show that our method outperforms the state-of-the-art algorithm on random k-CNF formulas and has effective application to approximate SSAT on circuit benchmarks.  ...  This paper focuses on random-exist quantified SSAT formulas, and proposes an algorithm combining binary decision diagram (BDD), logic synthesis, and modern SAT techniques to improve computational efficiency  ...  Random k-CNF Formulas The random k-CNF formulas are generated using the cnfgen command [Lauria, 2012] .  ... 
doi:10.24963/ijcai.2017/96 dblp:conf/ijcai/LeeWJ17 fatcat:lcgtk2zz75gqvcz2jwzzikvq3q

BIRD: Engineering an Efficient CNF-XOR SAT Solver and Its Applications to Approximate Model Counting

Mate Soos, Kuldeep S. Meel
2019 PROCEEDINGS OF THE THIRTIETH AAAI CONFERENCE ON ARTIFICIAL INTELLIGENCE AND THE TWENTY-EIGHTH INNOVATIVE APPLICATIONS OF ARTIFICIAL INTELLIGENCE CONFERENCE  
Given a Boolean formula φ, the problem of model counting, also referred to as #SAT is to compute the number of solutions of φ.  ...  The primary contribution of this paper is an affirmative answer to the above question. We present a novel architecture, called BIRD, to handle CNF-XOR formulas arising from hashingbased techniques.  ...  Acknowledgements We are grateful to the anonymous Reviewer #3 for the excellent suggestions to rewrite the Introduction.  ... 
doi:10.1609/aaai.v33i01.33011592 fatcat:jfq2fifo6vemxmggx5sagyukp4
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