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The Complexity of Satisfiability of Small Depth Circuits [chapter]

Chris Calabro, Russell Impagliazzo, Ramamohan Paturi
2009 Lecture Notes in Computer Science  
We show an improved randomized algorithm for the satisfiability problem for circuits of constant depth d and a linear number of gates cn: for each d and c, the running time is 2 (1−δ)n where the improvement  ...  The algorithm can be adjusted for use with Grover's algorithm to achieve a run time of 2 1−δ 2 n on a quantum computer.  ...  Thus, we are primarily left with the question of the complexity of the satisfiability problem for linear-size circuits if SETH holds.  ... 
doi:10.1007/978-3-642-11269-0_6 fatcat:wip2jerrsndgtjmbz2nzv26r7e

Connecting SAT Algorithms and Complexity Lower Bounds [chapter]

Ryan Williams
2011 Lecture Notes in Computer Science  
I will describe prior and current work on connecting the art of finding good satisfiability algorithms with the art of proving complexity lower bounds: proofs of limitations on what problems can be solved  ...  Surprisingly, even minor algorithmic progress on solving the circuit satisfiability problem faster than exhaustive search can be applied to prove strong circuit complexity lower bounds.  ...  It is anticipated that further progress in complexity lower bounds will be made by studying the complexity of satisfiability.  ... 
doi:10.1007/978-3-642-21581-0_1 fatcat:3bpt4mnhbbg5fjmpjtda5ymtca

A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits [article]

Russell Impagliazzo, Ramamohan Paturi, Stefan Schneider
2013 arXiv   pre-print
Our second motivation is to explore the connection between the expressive power of the circuits and the complexity of the corresponding circuit satisfiability problem.  ...  We believe that this is the first nontrivial satisfiability algorithm for cn-wire threshold circuits of depth two.  ...  threshold circuits of depth 2.  ... 
arXiv:1212.4548v2 fatcat:hqi6f7nz4bhvhcr3rcnx4i5wi4

A Satisfiability Algorithm for Sparse Depth Two Threshold Circuits

Russell Impagliazzo, Ramamohan Paturi, Stefan Schneider
2013 2013 IEEE 54th Annual Symposium on Foundations of Computer Science  
Our second motivation is to explore the connection between the expressive power of the circuits and the complexity of the corresponding circuit satisfiability problem.  ...  We give a nontrivial algorithm for the satisfiability problem for threshold circuits of depth two with a linear number of wires which improves over exhaustive search by an exponential factor.  ...  Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation  ... 
doi:10.1109/focs.2013.58 dblp:conf/focs/ImpagliazzoPS13 fatcat:6x3as7ngizcydac4mpgx7hybve

Deterministic simulation of probabilistic constant depth circuits

Miklos Ajtai, Avi Wigderson
1985 26th Annual Symposium on Foundations of Computer Science (sfcs 1985)  
Moreover, the functions $f_{n,\varepsilon}$ themselves can be computed by uniform polynomial size, constant depth circuits. Some (interrelated) consequences of this result are given below.  ...  property: for a random seed, the pseudorandom output "looks random" to any polynomial size, constant depth, unbounded fan-in circuit.  ...  We also present two results for the special case of depth 2 circuits. They deal, respectively, with finding an assignment and approximately counting the number of assignments.  ... 
doi:10.1109/sfcs.1985.19 dblp:conf/focs/AjtaiW85 fatcat:5qb2lcvhabhznmf7msgi262rga

On Medium-Uniformity and Circuit Lower Bounds

Rahul Santhanam, Ryan Williams
2013 2013 IEEE Conference on Computational Complexity  
We explore relationships between circuit complexity, the complexity of generating circuits, and algorithms for analyzing circuits. Our results can be divided into two parts: 1.  ...  of n O(1) size, output yes when C is unsatisfiable, and output no when C has at least 2 n−2 satisfying assignments.  ...  The circuits simulate C n on x implicitly, running the small-depth circuit for L succ to retrieve any bit of C n that is required.  ... 
doi:10.1109/ccc.2013.40 dblp:conf/coco/SanthanamW13 fatcat:mm4ls445rjd37mkuqv5sg2lqiy

Satisfiability and Derandomization for Small Polynomial Threshold Circuits

Valentine Kabanets, Zhenjian Lu, Michael Wagner
2018 International Workshop on Approximation Algorithms for Combinatorial Optimization  
We study the problems of exact and (promise) approximate counting for PTF circuits of constant depth. Satisfiability (#SAT ).  ...  This extends the recent result of Tell (STOC, 2018) for constant-depth LTF circuits of super-linear wire complexity. Pseudorandom generators.  ...  Acknowledgements We thank Suguru Tamaki for suggesting to us to consider sparse PTFs in the case of satisfiability, and for clarifying the MAX-k-SAT algorithm in [20] for us.  ... 
doi:10.4230/lipics.approx-random.2018.46 dblp:conf/approx/KabanetsL18 fatcat:ch4gewg3ybelzjjwutioqml6yq

Average-Case Lower Bounds and Satisfiability Algorithms for Small Threshold Circuits

Ruiwen Chen, Rahul Santhanam, Srikanth Srinivasan, Marc Herbstritt
2016 Computational Complexity Conference  
Siu, Roychowdhury, and Kailath [45] considered majority circuits of bounded depth and small gate complexity. They showed that Parity can be computed by depth-d circuits with O(dn 1/(d 1) ) gates.  ...  We give satisfiability algorithms beating brute force search for depth-d threshold circuits with a superlinear number of wires. These are the first such algorithms for depth greater than 2.  ...  Fact 35 (Small wire-complexity to small number of reads). Let C be any threshold circuit on n variables with wire complexity at most cn.  ... 
doi:10.4230/lipics.ccc.2016.1 dblp:conf/coco/ChenSS16 fatcat:ju424xsqsrhhnalgoo3uq4x7ti

On Uniformity and Circuit Lower Bounds

Rahul Santhanam, Ryan Williams
2014 Computational Complexity  
We explore relationships between circuit complexity, the complexity of generating circuits, and algorithms for analyzing circuits. Our results can be divided into two parts: 1.  ...  Our key insight is that if we assume that a "medium complexity" class has small medium-uniform circuits, this assumption can be applied in multiple ways: not only is it applicable to a language L in the  ...  Any opinions, findings and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the NSF.  ... 
doi:10.1007/s00037-014-0087-y fatcat:4fy5xs7jrfbtzm37apd7wq2qza

Algorithms for Circuits and Circuits for Algorithms

Ryan Williams
2014 2014 IEEE 29th Conference on Computational Complexity (CCC)  
The title of this paper is meant to highlight an emerging duality between two fundamental topics in algorithms and complexity theory.  ...  Algorithms for circuits refers to the design of interesting algorithms which can perform non-trivial circuit analysis of some kind, on either a circuit or a Boolean function given as a truth table.  ...  ACKNOWLEDGMENT The author acknowledges support by the Alfred P. Sloan foundation, Microsoft Research, and the NSF under grant CCF-1212372.  ... 
doi:10.1109/ccc.2014.33 dblp:conf/coco/Williams14 fatcat:t2irb2av2bcb5hzfi2dxkrsvqu

A Satisfiability Algorithm for Some Class of Dense Depth Two Threshold Circuits

Kazuyuki AMANO, Atsushi SAITO
2015 IEICE transactions on information and systems  
One of our motivations is to consider the relationship between the various circuit classes and the complexity of the corresponding circuit satisfiability problem of these classes.  ...  Recently, Impagliazzo et al. constructed a nontrivial algorithm for the satisfiability problem for sparse threshold circuits of depth two which is a class of circuits with cn wires.  ...  depth two circuits with small number of gates.  ... 
doi:10.1587/transinf.2014edp7127 fatcat:ocjj6xlifbasvee6rtghqrgene

On the Arithmetic Complexity of Euler Function [chapter]

Manindra Agrawal
2011 Lecture Notes in Computer Science  
The theorem can be strengthened to show that permanent polynomial requires size s(n) where s is any function satisfying s(s(n)) = 2 o(n) .  ...  The theorem can be strengthened to show that permanent polynomial requires size s(n) where s is any function satisfying s(s(n)) = 2 o(n) .  ...  Dedekind Eta Function η(z) = e πiz 12 E (e 2πiz ). η(z) is defined on the upper half of the complex plane and satisfies many interesting properties: η(z + 1) = e πi 12 η(z). η(− 1 z ) = √ −izη(z).  ... 
doi:10.1007/978-3-642-20712-9_4 fatcat:f7ghy3chpbfeviymw6wytmg6sy

Exponential Complexity of Satisfiability Testing for Linear-Size Boolean Formulas [chapter]

Evgeny Dantsin, Alexander Wolpert
2013 Lecture Notes in Computer Science  
The exponential complexity of the satisfiability problem for a given class of Boolean circuits is defined to be the infimum of constants α such that the problem can be solved in time poly(m) 2 αn , where  ...  The first one is about a similar "interweaving" between linearsize circuits of constant depth and k-CNFs [SS12].  ...  ; s ∞ = sup k {s k }; r d c is the exponential complexity of the satisfiability problem for circuits of depth at most d and size at most cn; r d ∞ = sup c {r d c }; f c is the exponential complexity of  ... 
doi:10.1007/978-3-642-38233-8_10 fatcat:ritdjkikvbfvzgic4aiymzqere

One-way functions and circuit complexity

R.B. Boppana, J.C. Lagarias
1987 Information and Computation  
We will show that the predicates Qi can be used to construct small circuits for g.  ...  RECEIVED November 1985; ACCEPTED February 17. 1987 circuit complexity of f by C(f) = min{size(C): C computes f }, and the circuit complexity of inverting such a function by C-'(f) = min{ C(g):g is an  ... 
doi:10.1016/0890-5401(87)90022-8 fatcat:b3p6uastine7nmhipvy4cwoqle

One-way functions and circuit complexity [chapter]

R. B. Boppana, J. C. Lagarias
1986 Lecture Notes in Computer Science  
We will show that the predicates Qi can be used to construct small circuits for g.  ...  RECEIVED November 1985; ACCEPTED February 17. 1987 circuit complexity of f by C(f) = min{size(C): C computes f }, and the circuit complexity of inverting such a function by C-'(f) = min{ C(g):g is an  ... 
doi:10.1007/3-540-16486-3_89 fatcat:2wixutunrjf2no27fku3cltg6m
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