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A Survey of Lower Bounds for Satisfiability and Related Problems
 Foundations and Trends in Theoretical Computer Science
, 2007
"... Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a lineartime, logarithmicspace algorithm for satisfiability was not ruled out. In 1997 Fortnow ..."
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Ever since the fundamental work of Cook from 1971, satisfiability has been recognized as a central problem in computational complexity. It is widely believed to be intractable, and yet till recently even a lineartime, logarithmicspace algorithm for satisfiability was not ruled out. In 1997 Fortnow, building on earlier work by Kannan, ruled out such an algorithm. Since then there has been a significant amount of progress giving nontrivial lower bounds on the computational complexity of satisfiability. In this article we survey the known lower bounds for the time and space complexity of satisfiability and closely related problems on deterministic, randomized, and quantum models with random access. We discuss the stateoftheart results and present the underlying arguments in a unified framework. 1
Algorithms and Resource Requirements for Fundamental Problems
, 2007
"... no. DGE0234630. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. ..."
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Cited by 10 (6 self)
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no. DGE0234630. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity.
A quantum timespace lower bound for the counting hierarchy
, 2007
"... We obtain the first nontrivial timespace lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real d and ..."
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We obtain the first nontrivial timespace lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real d and every positive real ǫ there exists a real c> 1 such that either: • MajMajSAT does not have a quantum algorithm with bounded twosided error that runs in time n c, or • MajSAT does not have a quantum algorithm with bounded twosided error that runs in time n d and space n 1−ǫ. In particular, MajMajSAT cannot be solved by a quantum algorithm with bounded twosided error running in time n 1+o(1) and space n 1−ǫ for any ǫ> 0. The key technical novelty is a time and spaceefficient simulation of quantum computations with intermediate measurements by probabilistic machines with unbounded error. We also develop a model that is particularly suitable for the study of general quantum computations with simultaneous time and space bounds. However, our arguments hold for any reasonable uniform model of quantum computation. 1
Automated proofs of time lower bounds
, 2007
"... A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6SAT, MajorityofMajoritySAT, and Tautologies, to name a few. The ..."
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A fertile area of recent research has demonstrated concrete polynomial time lower bounds for solving natural hard problems on restricted computational models. Among these problems are Satisfiability, Vertex Cover, Hamilton Path, MOD6SAT, MajorityofMajoritySAT, and Tautologies, to name a few. These lower bound proofs all follow a certain diagonalizationbased proofbycontradiction strategy. A pressing open problem has been to determine how powerful such proofs can possibly be. We propose an automated theoremproving methodology for studying these lower bound problems. In particular, we prove that the search for better lower bounds can often be turned into a problem of solving a large series of linear programming instances. We describe an implementation of a smallscale theorem prover and discover surprising experimental results. In some settings, our program provides strong evidence that the best known lower bound proofs are already optimal for the current framework, contradicting the consensus intuition; in others, the program guides us to improved lower bounds where none had been known for years.
A Status Report on the P versus NP Question
"... We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation. ..."
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We survey some of the history of the most famous open question in computing: the P versus NP question. We summarize some of the progress that has been made to date, and assess the current situation.
TimeSpace Efficient Simulations of Quantum Computations
, 2010
"... We give two time and spaceefficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations show that ev ..."
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We give two time and spaceefficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations show that every language solvable by a boundederror quantum algorithm running in time t and space s is also solvable by an unboundederror randomized algorithm running in time O(t · log t) and space O(s + log t), as well as by a boundederror quantum algorithm restricted to use an arbitrary universal set and running in time O(t · polylog t) and space O(s + log t), provided the universal set is closed under adjoint. We also develop a quantum model that is particularly suitable for the study of general computations with simultaneous time and space bounds. As an application of our randomized simulation, we obtain the first nontrivial lower bound for general quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are the problems of determining the truth value of a given Boolean formula whose variables are fully quantified by one or two majority quantifiers, respectively. We prove that for every real d and every positive real δ there exists a real c> 1 such that either • MajMajSAT does not have a boundederror quantum algorithm running in time O(n c), or • MajSAT does not have a boundederror quantum algorithm running in time O(n d) and space O(n 1−δ). In particular, MajMajSAT does not have a boundederror quantum algorithm running in time O(n 1+o(1) ) and space O(n 1−δ) for any δ> 0. Our lower bounds hold for any reasonable uniform model of quantum computation, in particular for the model we develop. 1
Applying practice to theory
 ACM SIGACT News
, 2008
"... How can complexity theory and algorithms benefit from practical advances in computing? We give a short overview of some prior work using practical computing to attack problems in computational complexity and algorithms, informally describe how linear program solvers may be used to help prove new low ..."
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How can complexity theory and algorithms benefit from practical advances in computing? We give a short overview of some prior work using practical computing to attack problems in computational complexity and algorithms, informally describe how linear program solvers may be used to help prove new lower bounds for satisfiability, and suggest a research program for developing new understanding in circuit complexity. 1
2012 Dieter van Melkebeek and Thomas Watson Licensed under a Creative Commons Attribution License
, 2011
"... Abstract: We give two time and spaceefficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations sh ..."
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Abstract: We give two time and spaceefficient simulations of quantum computations with intermediate measurements, one by classical randomized computations with unbounded error and the other by quantum computations that use an arbitrary fixed universal set of gates. Specifically, our simulations show that every language solvable by a boundederror quantum algorithm running in time t and space s is also solvable by an unboundederror randomized algorithm running in time O(t · logt) and space O(s + logt), as well as by a boundederror quantum algorithm restricted to use an arbitrary universal set and running in time O(t · polylogt) and space O(s + logt), provided the universal set is closed under adjoint. We also develop a quantum model that is particularly suitable for the study of general computations with simultaneous time and space bounds. As an application of our randomized simulation, we obtain the first nontrivial lower bound for general quantum algorithms solving problems related to satisfiability. Our bound applies to MAJSAT and MAJMAJSAT, which are the problems of determining the truth
unknown title
"... interest in STACS has remained at a high level over the past years. The STACS 2010 call for papers led to over 238 submissions from 40 countries. Each paper was assigned to three program committee members. The committee selected 54 papers during a two week electronic meeting held in November. As co ..."
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interest in STACS has remained at a high level over the past years. The STACS 2010 call for papers led to over 238 submissions from 40 countries. Each paper was assigned to three program committee members. The committee selected 54 papers during a two week electronic meeting held in November. As cochairs of the program committee, we would like to sincerely thank its members and the many external referees for their valuable work. In particular, there were intense and interesting discussions. The overall very high quality of the submissions made the selection a difficult task. We would like to express our thanks to the three invited speakers, Mikołaj Bojańczyk, Rolf Niedermeier, and Jacques Stern. Special thanks go to Andrei Voronkov for his EasyChair software (www.easychair.org). Moreover, we would like to warmly thank Wadie Guizani for preparing the conference proceedings and continuous help throughout the conference organization. For the third time, this year’s STACS proceedings are published in electronic form. A printed version was also available at the conference, with ISBN. The electronic proceedings are available through several portals, and in particular through HAL and LIPIcs series. The proceedings of the Symposium, which are published electronically in