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Exact enumeration of satisfiable 2-SAT formulae [article]

Sergey Dovgal and Élie de Panafieu and Vlady Ravelomanana
2021 arXiv   pre-print
We obtain exact expressions counting the satisfiable 2-SAT formulae and describe the structure of associated implication digraphs. Our approach is based on generating function manipulations.  ...  We expect these exact formulae to be amenable to rigorous asymptotic analysis using complex analytic tools, leading to a more detailed picture of the 2-SAT phase transition in the future.  ...  Having exact expressions for 2-SAT formulae potentially opens many new possibilities to describe the properties of a typical 2-SAT formula.  ... 
arXiv:2108.08067v1 fatcat:kxuoquwxz5gobldqg3zw5rtehy

MAX-SAT for Formulas with Constant Clause Density Can Be Solved Faster Than in $\mathcal{O}(2^n)$ Time [chapter]

Evgeny Dantsin, Alexander Wolpert
2006 Lecture Notes in Computer Science  
We give an exact deterministic algorithm for MAX-SAT.  ...  Worst-case upper bounds for MAX-SAT less than O(2 n ) were previously known only for k-CNF formulas and for CNF formulas with small clause density. where |F | is the size of input formula F , m is the  ...  There is an exact deterministic algorithm that solves MAX-SAT for formulas with constant clause density inO(c n ) time where c < 2.  ... 
doi:10.1007/11814948_26 fatcat:jybmdhppwfatvdp7tqn765wpgi

Enumerative counting is hard

Jin-Yi Cai, Lane A. Hemachandra
1989 Information and Computation  
An n-variable Boolean formula may have anywhere from 0 to 2" satisfying assignments.  ...  Can a polynomial-time machine, given such a formula, reduce this exponential number of possibilities to a small number of possibilities?  ...  We have r new leaves, where 26r=2 ISI <2k. 2. COMBINE FORMULAS and RUN ENUMERATOR. Combine f  ... 
doi:10.1016/0890-5401(89)90063-1 fatcat:bgvpotvtqnchxheic32ny77hpe

Enumerative counting is hard

J.-Y. Cai, L.A. Hemachandra
1988 [1988] Proceedings. Structure in Complexity Theory Third Annual Conference  
An n-variable Boolean formula may have anywhere from 0 to 2" satisfying assignments.  ...  Can a polynomial-time machine, given such a formula, reduce this exponential number of possibilities to a small number of possibilities?  ...  We have r new leaves, where 26r=2 ISI <2k. 2. COMBINE FORMULAS and RUN ENUMERATOR. Combine f  ... 
doi:10.1109/sct.1988.5279 dblp:conf/coco/CaiH88 fatcat:mzratsooqjg5riutgc6x2inpua

Satisfiability on Mixed Instances

Ruiwen Chen, Rahul Santhanam
2016 Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science - ITCS '16  
We show that there is an algorithm which for any such formula with a constant number of quantifier blocks and of size n c , where c < 5/4, solves satisfiability in time 2 n−n Ω(1) .  ...  The study of the worst-case complexity of the Boolean Satisfiability (SAT) problem has seen considerable progress in recent years, for various types of instances including CNFs [16, 15, 20, 21] , Boolean  ...  Introduction Boolean Satisfiability (SAT) is the canonical NP-complete problem. Much effort has gone into designing and analyzing exact algorithms for SAT.  ... 
doi:10.1145/2840728.2840768 dblp:conf/innovations/ChenS16 fatcat:pkhlcnd75beernitffblimhuvm

Smallest MUS Extraction with Minimal Hitting Set Dualization [chapter]

Alexey Ignatiev, Alessandro Previti, Mark Liffiton, Joao Marques-Silva
2015 Lecture Notes in Computer Science  
An unsatisfiable formula can have multiple MUSes, some of which provide more insights than others. Different criteria can be considered in order to identify a good minimal explanation.  ...  Among these, the size of an MUS is arguably one of the most intuitive.  ...  Call a SAT solver on the resulting formula, requesting an MHS of size k. If the formula is satisfiable, then we have a new MHS of size k that also hits C.  ... 
doi:10.1007/978-3-319-23219-5_13 fatcat:q2qlgjswybe2fnypjqnczktyty

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.  ...  Using this method, we show the first known results on the density of states of several widely used formulas and we provide novel insights about the behavior of random 3-SAT formulas around the phase transition  ...  The true density is computed by exact enumeration.  ... 
doi:10.1007/978-3-642-15396-9_6 fatcat:ba4b72rtbzayhg2bc3bfugse5i

Paradigms for Parameterized Enumeration [article]

Nadia Creignou and Arne Meier and Julian-Steffen Müller and Johannes Schmidt and Heribert Vollmer
2013 arXiv   pre-print
First we define formally different notions of efficient enumeration in the context of parameterized complexity.  ...  The aim of the paper is to examine the computational complexity and algorithmics of enumeration, the task to output all solutions of a given problem, from the point of view of parameterized complexity.  ...  Enumeration classification for MaxOnes-SAT The self-reducibility technique was in particular applied in order to enumerate all satisfying assignments of a generalized CNF-formula [1] , thus allowing to  ... 
arXiv:1306.2171v1 fatcat:632mpnrbofcstbad5plje2vle4

An Exact Algorithm for the Boolean Connectivity Problem for k-CNF [chapter]

Kazuhisa Makino, Suguru Tamaki, Masaki Yamamoto
2010 Lecture Notes in Computer Science  
We present an exact algorithm for a PSPACE-complete problem, denoted by CONNkSAT, which asks if the solution space for a given k-CNF formula is connected on the n-dimensional hypercube.  ...  that the SAT problem (with no restriction to the clause length) is not solvable in time O((2 − ) n ) for any constant > 0.  ...  number of vertices of a graph for the traveling salesman problem, or the number of variables of a formula for the satisfiability problem) is there an exact algorithm for the problem in time O(2 n ), or  ... 
doi:10.1007/978-3-642-14186-7_15 fatcat:uywyeojbbbgnfinv7zkgae57ca

An exact algorithm for the Boolean connectivity problem for k-CNF

Kazuhisa Makino, Suguru Tamaki, Masaki Yamamoto
2011 Theoretical Computer Science  
We present an exact algorithm for a PSPACE-complete problem, denoted by CONNkSAT, which asks whether the solution space for a given k-CNF formula is connected on the ndimensional hypercube.  ...  > 0, provided that the SAT problem (with no restriction to the clause length) is not solvable in time O((2 − ϵ) n ) for any constant ϵ > 0.  ...  number of vertices of a graph for the traveling salesman problem, or the number of variables of a formula for the satisfiability problem) is there an exact algorithm for the problem in time O(2 n ), or  ... 
doi:10.1016/j.tcs.2011.04.041 fatcat:jc65muc7cvfcfnstkind7uc4mu

An Approximation Algorithm for #k-SAT

Marc Thurley, Marc Herbstritt
2012 Symposium on Theoretical Aspects of Computer Science  
We present a simple randomized algorithm that approximates the number of satisfying assignments of Boolean formulas in conjunctive normal form.  ...  fastest exact algorithms for the problem.  ...  Formulas with few solutions For formulas with few solutions we will now present an algorithm relying on a simple enumeration of solutions by using a k-SAT algorithm as a subroutine.  ... 
doi:10.4230/lipics.stacs.2012.78 dblp:conf/stacs/Thurley12 fatcat:375aoaw36nclvckyjuip4bv4kq

Generating functions and the satisfiability threshold

Vincent Puyhaubert
2004 Discrete Mathematics & Theoretical Computer Science  
Although the threshold has been proved to exist for the 2-SAT formulæ and for closely related problems like 3-XORSAT, there is still no proof for the 3-sat problem.  ...  International audience The 3-SAT problem consists in determining if a boolean formula with 3 literals per clause is satisfiable.  ...  Acknowledgements I want to thank Olivier Dubois for all the time he took to explain his articles to me, and Philippe Flajolet for pointing to generating functions as a useful tool for enumeration in threshold  ... 
doi:10.46298/dmtcs.325 fatcat:tes7p725vncadciwxncc75mu6i

An Approximation Algorithm for #k-SAT [article]

Marc Thurley
2011 arXiv   pre-print
We present a simple randomized algorithm that approximates the number of satisfying assignments of Boolean formulas in conjunctive normal form.  ...  fastest exact algorithms for the problem.  ...  Formulas with few solutions For formulas with few solutions we will now present an algorithm relying on a simple enumeration of solutions by using a k-SAT algorithm as a subroutine.  ... 
arXiv:1107.2001v1 fatcat:4ti42kporvef7kgmm2qhlqotli

Towards backbone computing: A Greedy-Whitening based approach

Yueling Zhang, Min Zhang, Geguang Pu, Fu Song, Jianwen Li, Pascal Fontaine, Cezary Kaliszyk, Stephan Schulz, Josef Urban
2018 AI Communications  
Computing a part of backbone efficiently could guide the following searching in SAT solving and accelerate the process, which is widely used in fault localization, product configuration, and formula simplification  ...  Backbone computing is accelerated since SAT testings of literals in BL ( ) are saved.  ...  Acknowledgements This work is supported the by the program of Theoretical and Practical Research about Linear Time Logic (LTL) Satisfiability (61572197) from National Natural Science Foundation of China  ... 
doi:10.3233/aic-180763 fatcat:tjrrzv7hvne5hbs2v6d54ipcbq

Exact sat-based toffoli network synthesis

Daniel Große, Xiaobo Chen, Gerhard W. Dueck, Rolf Drechsler
2007 Proceedings of the 17th great lakes symposium on Great lakes symposium on VLSI - GLSVLSI '07  
Our iterative algorithm formulates the synthesis problem with d Toffoli gates as a sequence of Boolean Satisfiability (SAT) instances.  ...  In this paper, we present the first exact synthesis algorithm for reversible functions using generalized Toffoli gates.  ...  In this paper we present the first exact algorithm for Toffoli synthesis of a reversible function 1 . Our method uses an iterative algorithm that is based on Boolean Satisfiability (SAT).  ... 
doi:10.1145/1228784.1228812 dblp:conf/glvlsi/GrosseCDD07 fatcat:mm3iiizrn5dz3glfrbjh4q52cu
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