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An improved exponential-time algorithm for k-SAT

Ramamohan Paturi, Pavel Pudlák, Michael E. Saks, Francis Zane
2005 Journal of the ACM  
In addition, it is the fastest known probabilistic algorithm for k-CNF, k ≥ 3, that have at most one satisfying assignment (unique k-SAT) (with a running time O(2 (2 ln 2−1)n+o(n) ) = O(2 0.386...n ) in  ...  Currently, this is the fastest known probabilistic algorithm for k-CNF satisfiability for k ≥ 4 (with a running time of O(2 0.5625n ) for 4-CNF).  ...  I THE EXPONENT c IN THE BOUND 2 cn−o(n) OF OUR ALGORITHM FOR UNIQUE-k-SAT, FOR k-SAT AND THE CORRESPONDING BOUNDS FOR SCHÖNING'S ALGORITHM SCHÖNING [1999] AND ITS IMPROVED VERSION FOR 3-SAT [HOFMEISTER  ... 
doi:10.1145/1066100.1066101 fatcat:tyjmcqj32jhpvfb7o7syyjjyvm

An improved exponential-time algorithm for k-SAT

R. Paturi, P. Pudlik, M.E. Saks, F. Zane
Proceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)  
For each , the bounds for general -CNF are the best known for the worst-case complexity of finding a satisfying solution for - SAT.  ...  In particular, we show that the algorithm finds a satisfying assignment of a general 3-CNF in time Ç´¾ Ò µ with high probability; where the best previous algorithm [9, 10] has running time Ç´¾ ¾Ò µ.  ...  Acknowledgments: We thank Professor Ed Bender for his help in evaluation the integral in Section 3.4. We also thank Yuming Zhang for his helpful comments and corrections.  ... 
doi:10.1109/sfcs.1998.743513 dblp:conf/focs/PaturiPSZ98 fatcat:aags4tulljbhheba7da4wwg2ce

Improved exact algorithms for mildly sparse instances of Max SAT

Takayuki Sakai, Kazuhisa Seto, Suguru Tamaki, Junichi Teruyama
2017 Theoretical Computer Science  
We present improved exponential time exact algorithms for Max SAT. Our algorithms run in time of the form O(2 (1−µ(c))n ) for instances with n variables and m = cn clauses.  ...  In this setting, there are three incomparable currently best algorithms: a deterministic exponential space algorithm with µ(c) = O((c log c) 2/3 ) and for Max 3-SAT with µ(c) = 1 O(c 1/2 ) that run super-polynomially  ...  We would like to thank anonymous reviewers for their helpful comments on our paper.  ... 
doi:10.1016/j.tcs.2017.07.011 fatcat:krzk4dqwyzbxhhgzebfqu2f7t4

Improved Algorithms for Sparse MAX-SAT and MAX-k-CSP [chapter]

Ruiwen Chen, Rahul Santhanam
2015 Lecture Notes in Computer Science  
We also give an algorithm with improved savings for satisfiability of depth-2 threshold circuits with cn wires.  ...  For instances with n variables and cn clauses (constraints), we give algorithms running in time poly(n)· 2 n(1−µ) for µ = Ω( 1 c ) and polynomial space solving MAX-SAT and MAX-k-SAT, µ = Ω( 1 √ c ) and  ...  An exponential-space algorithm for MAX-k-CSP We next give an algorithm with improved running time but using exponential space.  ... 
doi:10.1007/978-3-319-24318-4_4 fatcat:vcvuiznka5esbku2vktkt7qn7u

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  
Improving prior reductions, we show the time complexities of Unique k-SAT and k-SAT are very tightly related: if Unique k-SAT is in 2n(1−f(k)/k) time for an unbounded f, then k-SAT is in 2n(1−f(k)(1−ɛ)  ...  It is natural to hypothesize that the worst-case (exponential-time) complexity of Unique k-SAT is substantially less than that of k-SAT.  ...  An algorithm running in 2 (1−f (k)/k)n time for Unique k-SAT (where f (k) is unbounded) implies a 2 (1−f (k)/k+O((log f (k))/k))n time algorithm for k-SAT.  ... 
doi:10.1609/aaai.v34i09.7125 fatcat:jocm2ycqdrcobed5yopysjdotm

Exact Algorithms and Complexity [chapter]

Ramamohan Paturi
2010 Lecture Notes in Computer Science  
Over the past couple of decades, a series of exact exponential-time algorithms have been developed with improved run times for a number of problems including IndependentSet, k-SAT, and k-colorability using  ...  This class includes Davis-Putnam-style backtracking algorithms developed in recent times to provide improved exponential-time upper bounds for a variety of NP-hard problems.  ...  Over the past couple of decades, a series of exact exponential-time algorithms have been developed with improved run times for a number of problems including IndependentSet, k-SAT, and k-colorability using  ... 
doi:10.1007/978-3-642-14186-7_2 fatcat:mfj4hew47jfynlwykzntmukirm

An Improved Exact Algorithm for the Domatic Number Problem [article]

Tobias Riege, Jörg Rothe, Holger Spakowski, Masaki Yamamoto
2006 arXiv   pre-print
To prove our result, we combine an algorithm by Fomin et al. with Yamamoto's algorithm for the satisfiability problem.  ...  In addition, we show that the 3-domatic number problem can be solved for graphs G with bounded maximum degree Delta(G) by a randomized algorithm, whose running time is better than the previous bound due  ...  We are grateful to Osamu Watanabe and Dieter Kratsch for inspiring discussions on the subject of this paper. In particular, we thank Dieter Kratsch for calling our attention to Lawler's algorithm.  ... 
arXiv:cs/0603060v1 fatcat:tthcokzv7fe53h4gwwasry3jky

Relaxed Random Search for Solving K-Satisfiability and its Information Theoretic Interpretation

Amirahmad Nayyeri, Gholamhossein Dastghaibyfard
2017 International Journal of Advanced Computer Science and Applications  
complexity of using randomized algorithm for finding the solution of K-SAT in more relaxed regions.  ...  In this paper, by considering the recent approach of applying statistical physic methods for analyzing the phase transition in the complexity of algorithms used for solving K-SAT, we try to compute the  ...  In spite of all efforts, up to now, we don't have a polynomial time algorithm for solving K-SAT.  ... 
doi:10.14569/ijacsa.2017.081153 fatcat:lovp3rn35bfbdk3iemvl3qwy7y

Approximating MAX SAT by moderately exponential and parameterized algorithms

Bruno Escoffier, Vangelis Th. Paschos, Emeric Tourniaire
2014 Theoretical Computer Science  
We study approximation of the max sat problem by moderately exponential algorithms.  ...  than those needed for exact computation, approximation ratios unachievable in polynomial time.  ...  Note that finding an exact algorithm in time O * (γ n ) for some γ < 2 is a famous open question for max sat (cf. the strong exponential time hypothesis [21] ) as well as for some other combinatorial  ... 
doi:10.1016/j.tcs.2014.10.039 fatcat:p4spt2lilzgb3kgxqwzbws3ncm

On the Exact Complexity of Evaluating Quantified k -CNF

Chris Calabro, Russell Impagliazzo, Ramamohan Paturi
2012 Algorithmica  
On the other hand, a nontrivial exponential-time algorithm for Π23-sat would provide a k-sat solver with better exponent than all current algorithms for sufficiently large k.  ...  Therefore, if we assume the Strong Exponential-Time Hypothesis, then there is no algorithm for Π23-sat running in time 2 cn with c < 1.  ...  We thank the reviewers for their helpful comments. Steering Committee Hans  ... 
doi:10.1007/s00453-012-9648-0 fatcat:pqo7syfxsre4znksn3ds5ixwlu

Worst-case study of local search for MAX-k-SAT

Edward A. Hirsch
2003 Discrete Applied Mathematics  
Our algorithm and its analysis are based on Sch oning's proof of the best current worst-case time bound for k-SAT (in: algorithm makes random walks of polynomial length.  ...  In this paper, we give a randomized (1 − )-approximation algorithm for MAX-k-SAT whose worst-case time bound depends on the number of variables.  ...  Acknowledgements The useful comments of three anonymous referees of Discrete Applied Mathematics greatly improved the presentation of this paper and added the last open question.  ... 
doi:10.1016/s0166-218x(02)00404-3 fatcat:m3cj3kmpgjd33nhfk3v7kjvmnu

On the possibility of faster SAT algorithms [chapter]

Mihai Pătraşcu, Ryan Williams
2010 Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms  
We show that attaining any of the following bounds would improve the state of the art in algorithms for SAT: • an O(n k−ε ) algorithm for k-Dominating Set, for any k ≥ 3, • a (computationally efficient  ...  length, and • an O((n + m) k−ε ) algorithm for HornSat with k unrestricted length clauses.  ...  R.W. thanks the organizers of Dagstuhl Seminar 05301 on Exact Algorithms and Fixed-Parameter Tractability for their invitation which initiated this work, and his thesis committee for their feedback on  ... 
doi:10.1137/1.9781611973075.86 dblp:conf/soda/PatrascuW10 fatcat:lomltgvirfahxo34h25wbqatkq

Approximating MAX SAT by Moderately Exponential and Parameterized Algorithms [chapter]

Bruno Escoffier, Vangelis Th. Paschos, Emeric Tourniaire
2012 Lecture Notes in Computer Science  
We study approximation of the max sat problem by moderately exponential algorithms.  ...  than those needed for exact computation, approximation ratios unachievable in polynomial time.  ...  Note that finding an exact algorithm in time O * (γ n ) for some γ < 2 is a famous open question for max sat (cf. the strong exponential time hypothesis [21] ) as well as for some other combinatorial  ... 
doi:10.1007/978-3-642-29952-0_23 fatcat:pzn74vb56ze6xjo2c2psh23eqq

An Approximation Algorithm for #k-SAT

Marc Thurley, Marc Herbstritt
2012 Symposium on Theoretical Aspects of Computer Science  
To the best of our knowledge this is the first algorithm which approximates #k-SAT for any k ≥ 3 within a running time that is not only non-trivial, but also significantly better than that of the currently  ...  fastest exact algorithms for the problem.  ...  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

An Approximation Algorithm for #k-SAT [article]

Marc Thurley
2011 arXiv   pre-print
To the best of our knowledge this is the first algorithm which approximates #k-SAT for any k >= 3 within a running time that is not only non-trivial, but also significantly better than that of the currently  ...  fastest exact algorithms for the problem.  ...  The cases k = 3 and k = 4. For these values of k, several improvements over the PPSZ algorithm have been presented.  ... 
arXiv:1107.2001v1 fatcat:4ti42kporvef7kgmm2qhlqotli
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