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DNF sparsification and a faster deterministic counting algorithm

Parikshit Gopalan, Raghu Meka, Omer Reingold
2013 Computational Complexity  
We give a faster deterministic algorithm for approximately counting the number of satisfying solutions to a DNF or CNF.  ...  Given a DNF(or CNF) f on n variables and poly(n) terms, we give a deterministic nÕ ((log log n) 2 ) time algorithm that computes an (additive) ε approximation to the fraction of satisfying assignments  ...  ACKNOWLEDGEMENTS We thank Adam Klivans, Rocco Servedio, Avi Wigderson and David Zuckerman for valuable discussions. We thank Rocco for drawing our attention to Friedgut's theorem in this context.  ... 
doi:10.1007/s00037-013-0068-6 fatcat:2jwkc47lv5cy3ciwnezyg2arpa

DNF Sparsification and a Faster Deterministic Counting Algorithm [article]

Parikshit Gopala, Raghu Meka, Omer Reingold
2012 arXiv   pre-print
We combine our sparsification result with the work of Luby and Velikovic to give a faster deterministic algorithm for approximately counting the number of satisfying solutions to a DNF.  ...  Given a DNF formula on n variables, the two natural size measures are the number of terms or size s(f), and the maximum width of a term w(f).  ...  Acknowledgements We thank Adam Klivans, Ryan O'Donnell, Rocco Servedio, Avi Wigderson and David Zuckerman for valuable discussions.  ... 
arXiv:1205.3534v1 fatcat:e23ypu6yafgzni35gvz6eqzqgi

DNF Sparsification and a Faster Deterministic Counting Algorithm

Parikshit Gopalan, Raghu Meka, Omer Reingold
2012 2012 IEEE 27th Conference on Computational Complexity  
We give a faster deterministic algorithm for approximately counting the number of satisfying solutions to a DNF or CNF.  ...  Given a DNF(or CNF) f on n variables and poly(n) terms, we give a deterministic nÕ ((log log n) 2 ) time algorithm that computes an (additive) ε approximation to the fraction of satisfying assignments  ...  ACKNOWLEDGEMENTS We thank Adam Klivans, Rocco Servedio, Avi Wigderson and David Zuckerman for valuable discussions. We thank Rocco for drawing our attention to Friedgut's theorem in this context.  ... 
doi:10.1109/ccc.2012.38 dblp:conf/coco/GopalanMR12 fatcat:zavqxtjwcbaqjlzswy4guu7zzi

Special Issue "Conference on Computational Complexity 2012" Guest editors' foreword

Boaz Barak, Irit Dinur
2013 Computational Complexity  
On sunflowers and matrix multiplication  ...  These papers were selected for invitation by the program committee from all the papers presented at the conference, and following submission underwent the standard refereeing process.  ...  DNF sparsification and a faster deterministic counting algo One of the hallmarks of computational complexity has been the interplay between algorithms and lower bounds.  ... 
doi:10.1007/s00037-013-0070-z fatcat:nosn2cjygrcixofcfvqonshuni

Counting Solutions to Polynomial Systems via Reductions

R. Ryan Williams, Marc Herbstritt
2018 ACM-SIAM Symposium on Discrete Algorithms  
The approximate-counting algorithm yields efficient approximate counting for other well-known problems, such as 2-SAT, NAE-3SAT, and 3-Coloring.  ...  We apply this reduction to give a deterministic poly(n, s, log p)/ε 2 -time algorithm for approximately counting the fraction of solutions to a system of s quadratic n-variate polynomials over F p (the  ...  There, I got a chance to think more carefully about the ideas in the approximate counting algorithm.  ... 
doi:10.4230/oasics.sosa.2018.6 dblp:conf/soda/Williams18a fatcat:ojjepy2sijd47b2wtzfqrksrtu

Fighting Perebor: New and Improved Algorithms for Formula and QBF Satisfiability

Rahul Santhanam
2010 2010 IEEE 51st Annual Symposium on Foundations of Computer Science  
We investigate the possibility of finding satisfying assignments to Boolean formulae and testing validity of quantified Boolean formulae (QBF) asymptotically faster than a brute force search.  ...  Our second main result is a deterministic algorithm running in time 2 n−Ω(n/ log(n)) for solving QBFs in which the number of occurrences of any variable is bounded by a constant.  ...  For any integer k and structured set S of instances, there is a deterministic algorithm solving k-bounded TQBF which runs in time 2 n−Ω(n) on S.  ... 
doi:10.1109/focs.2010.25 dblp:conf/focs/Santhanam10 fatcat:kwv6bp73jfe6dmm5rn3byo324a

The Complexity of Propositional Proofs

Nathan Segerlind
2007 Bulletin of Symbolic Logic  
Questions about proof sizes have connections with computational complexity, theories of arithmetic, and satisfiability algorithms.  ...  This is article includes a broad survey of the field, and a technical exposition of some recently developed techniques for proving lower bounds on proof sizes.  ...  Thanks also go to Sam Buss for a helpful conversation on I∆ 0 , and Jakob Nordström for providing comments on an early draft.  ... 
doi:10.2178/bsl/1203350879 fatcat:mzffx4yqzfcg7ell47v452f2eu

How many qubits are needed for quantum computational supremacy?

Alexander M. Dalzell, Aram W. Harrow, Dax Enshan Koh, Rolando L. La Placa
2020 Quantum  
an n×n integer-valued matrix, and assert that any non-deterministic algorithm that solves them requires 2cn time steps, where c∈{a,b}.  ...  Our first two conjectures poly3-NSETH(a) and per-int-NSETH(b) take specific classical counting problems related to the number of zeros of a degree-3 polynomial in n variables over F2 or the permanent of  ...  Finally, we acknowledge Richard Kueng for suggesting the proof strategy in Lemma 9, as well as Tomoyuki Morimae and Suguru Tamaki for comments on a draft of this paper.  ... 
doi:10.22331/q-2020-05-11-264 fatcat:6sjufj3vlbe3zeehesaf2v7gi4

How many qubits are needed for quantum computational supremacy? [article]

Alexander M. Dalzell, Aram W. Harrow, Dax Enshan Koh, Rolando L. La Placa
2020 arXiv   pre-print
Each version is parameterized by a constant a and asserts that certain specific computational problems with input size n require 2^an time steps to be solved by a non-deterministic algorithm.  ...  with 208 qubits, Quantum Approximate Optimization Algorithm (QAOA) circuits with 420 qubits and boson sampling circuits (i.e. linear optical networks) with 98 photons are large enough for the task of  ...  Finally, we acknowledge Richard Kueng for suggesting the proof strategy in Lemma 9, as well as Tomoyuki Morimae and Suguru Tamaki for comments on a draft of this paper.  ... 
arXiv:1805.05224v3 fatcat:vaypbyiqerao7n4ym7owfvxy3e

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

Kazuyuki AMANO, Atsushi SAITO
2015 IEICE transactions on information and systems  
We construct a nontrivial algorithm for a larger class of circuits.  ...  We give a nontrivial circuit satisfiability algorithm for a class of circuits which may not be sparse in gates with dependency.  ...  In what follows, we give a way to obtain Theorem 3.1 from Lemma 3.1. A way to get a deterministic algorithm from a two sided error algorithm with error probability at most 1/3 is given as follows.  ... 
doi:10.1587/transinf.2014edp7127 fatcat:ocjj6xlifbasvee6rtghqrgene

The Quantum Strong Exponential-Time Hypothesis [article]

Harry Buhrman, Subhasree Patro, Florian Speelman
2019 arXiv   pre-print
gap between verifier and prover.  ...  This hypothesis and its variants gave rise to a fruitful field of research, fine-grained complexity, obtaining (mostly tight) lower bounds for many problems in P whose unconditional lower bounds are hard  ...  HB, SP, and FS are additionally supported by NWO Gravitation grants NETWORKS and QSC, and EU grant QuantAlgo.  ... 
arXiv:1911.05686v2 fatcat:5qjdfoboofhjbo7c2n6jzvpxji

On Moderately Exponential Time for SAT [chapter]

Evgeny Dantsin, Alexander Wolpert
2010 Lecture Notes in Computer Science  
., in time p(|F |) 2 cn for some polynomial p and some constant c < 1, where F is a CNF formula of size |F | over n variables? This challenging question is far from being resolved.  ...  The extension is obtained by allowing a ∀ . . . ∀ ∃ . . . ∃ prefix in the first-order part.  ...  Acknowledgment We are grateful to the anonymous referees for their careful analysis and valuable suggestions. We implemented most of the suggestions in the final version.  ... 
doi:10.1007/978-3-642-14186-7_27 fatcat:vwu4po4v7jac7n4ifktf3wqctu

Quantum Computing: Lecture Notes [article]

Ronald de Wolf
2022 arXiv   pre-print
The first 10 chapters cover the circuit model and the main quantum algorithms (Deutsch-Jozsa, Simon, Shor, Hidden Subgroup Problem, Grover, quantum walks, Hamiltonian simulation and HHL).  ...  Appendices A and B give a brief introduction to the required linear algebra and some other mathematical and computer science background.  ...  One can think of a randomized algorithm as a probability distribution over deterministic algorithms (one deterministic algorithm for each setting of the coins). • When we say a (randomized or quantum)  ... 
arXiv:1907.09415v4 fatcat:c6g6boiwqzhglbzkkmjlt65upe

Improved Bounds for Quantified Derandomization of Constant-Depth Circuits and Polynomials * †

O' Ryan, Donnell
48 Leibniz International Proceedings in Informatics Schloss Dagstuhl-Leibniz-Zentrum für Informatik   unpublished
For constant-depth circuits, we construct an algorithm for quantified derandomization that works for a parameter B(n) that is only slightly smaller than a "threshold" parameter, and is significantly faster  ...  Intuitively, a standard technique to deterministically find a "good" object is to construct a simple deterministic test that decides the set of good objects, and then "fool" that test using a pseudorandom  ...  The author thanks his advisor, Oded Goldreich, for many helpful discussions, and for his guidance and support during the research and writing process.  ... 
fatcat:44jnfppkujal5ik6q2dfui5s2e

st Symposium on Simplicity in Algorithms New Orleans LA, USA Co-located with The 29th ACM-SIAM Symposium on Discrete Algorithms (SODA 2018)

Raimund Seidel
2018 unpublished
Ravi and Madhur Tulsiani for helpful discussions and comments on the presentation. Acknowledgement.  ...  We are very grateful to Dana Ron and Oded Goldreich for valuable discussions and comments. Acknowledgements. We thank the anonymous reviewers, especially reviewer 2, for their Acknowledgments.  ...  SIAM, 2017. doi:10.1137/1.9781611974782.143. 28 Michael Luby and Boban Velickovic. On deterministic approximation of DNF.  ... 
fatcat:tvuaojfsvjgfjaj7en2x4wdbhy
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