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16
On approximate majority and probabilistic time
 in Proceedings of the 22nd IEEE Conference on Computational Complexity
, 2007
"... We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and ..."
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Cited by 18 (6 self)
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We prove new results on the circuit complexity of Approximate Majority, which is the problem of computing Majority of a given bit string whose fraction of 1’s is bounded away from 1/2 (by a constant). We then apply these results to obtain new relationships between probabilistic time, BPTime (t), and alternating time, Σ O(1)Time (t). Our main results are the following: 1. We prove that 2 n0.1�size depth3 circuits for Approximate Majority on n bits have bottom fanin Ω(log n). As a corollary we obtain that BPTime (t) �⊆ Σ2Time � o(t 2) � with respect to some oracle. This complements the result that BPTime (t) ⊆ Σ2Time � t 2 · poly log t � with respect to every oracle (Sipser and Gács, STOC ’83; Lautemann, IPL ’83). 2. We prove that Approximate Majority is computable by uniform polynomialsize circuits of depth 3. Prior to our work, the only known polynomialsize depth3 circuits for Approximate Majority were nonuniform (Ajtai, Ann. Pure Appl. Logic ’83). We also prove that BPTime (t) ⊆ Σ3Time (t · poly log t). This complements our results in (1). 3. We prove new lower bounds for solving QSAT 3 ∈ Σ3Time (n · poly log n) on probabilistic computational models. In particular, we prove that solving QSAT 3 requires time n 1+Ω(1) on Turing machines with a randomaccess input tape and a sequentialaccess work tape that is initialized with random bits. No lower bound was previously known on this model (for a function computable in linear space). ∗ Author supported by NSF grant CCR0324906. Any opinions, findings and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the
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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Cited by 14 (1 self)
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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
TimeSpace Tradeoffs for Counting NP Solutions Modulo Integers
 In Proceedings of the 22nd IEEE Conference on Computational Complexity
, 2007
"... We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km sat ..."
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Cited by 12 (5 self)
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We prove the first timespace tradeoffs for counting the number of solutions to an NP problem modulo small integers, and also improve upon known timespace tradeoffs for Sat. Let m> 0 be an integer, and define MODmSat to be the problem of determining if a given Boolean formula has exactly km satisfying assignments, for some integer k. We show for all primes p except for possibly one of them, and for all c < 2cos(π/7) ≈ 1.801, there is a d> 0 such that MODpSat is not solvable in n c time and n d space by general algorithms. That is, there is at most one prime p that does not satisfy the tradeoff. We prove that the same limitation holds for Sat and MOD6Sat, as well as MODmSat for any composite m that is not a prime power. Our main tool is a general method for rapidly simulating deterministic computations with restricted space, by counting the number of solutions to NP predicates modulo integers. The simulation converts an ordinary algorithm into a “canonical ” one that consumes roughly the same amount of time and space, yet canonical algorithms have nice properties suitable for counting.
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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Cited by 5 (1 self)
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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
A New TimeSpace Lower Bound for Nondeterministic Algorithms Solving Tautologies
, 2007
"... We show that for all reals c and d such that c 2 d < 4 there exists a positive real e such that tautologies cannot be decided by both a nondeterministic algorithm that runs in time n c, and a nondeterministic algorithm that runs in time n d and space n e. In particular, for every real d < 3 √ ..."
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Cited by 3 (2 self)
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We show that for all reals c and d such that c 2 d < 4 there exists a positive real e such that tautologies cannot be decided by both a nondeterministic algorithm that runs in time n c, and a nondeterministic algorithm that runs in time n d and space n e. In particular, for every real d < 3 √ 4 there exists a positive real e such that tautologies cannot be decided by a nondeterministic algorithm that runs in time n d and space n e. 1
AlternationTrading Proofs, Linear Programming, and Lower Bounds
"... 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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Cited by 3 (2 self)
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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. The proofs of these lower bounds follow a certain proofbycontradiction strategy, which we call “resourcetrading” or “alternationtrading.” An important open problem is to determine how powerful such proofs can possibly be. We propose a methodology for studying these proofs that makes them amenable to both formal analysis and automated theorem proving. Formalizing the framework, 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 implement a smallscale theorem prover and report surprising results, which allow us to extract new humanreadable time lower bounds for several problems. We also use the framework to prove concrete limitations on the current techniques.
On probabilistic time versus alternating time
 Electronic Colloquium on Computational Complexity
, 2005
"... We prove several new results regarding the relationship between probabilistic time, BPTime(t), and alternating time, Σ O(1)Time(t). Our main results are the following: 1. We prove that BPTime(t) ⊆ Σ3Time(t · poly log t). Previous results show that BPTime (t) ⊆ Σ2Time � t 2 · log t � (Sipser and Gá ..."
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Cited by 3 (0 self)
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We prove several new results regarding the relationship between probabilistic time, BPTime(t), and alternating time, Σ O(1)Time(t). Our main results are the following: 1. We prove that BPTime(t) ⊆ Σ3Time(t · poly log t). Previous results show that BPTime (t) ⊆ Σ2Time � t 2 · log t � (Sipser and Gács, STOC ’83; Lautemann, IPL ’83) and BPTime(t) ⊆ ΣcTime(t) for a large constant c> 3 (Ajtai, Adv. in Comp. Complexity Theory ’93). 2. We prove that BPTime(t) � ⊆ Σ2Time � o(t 2) � with respect to some oracle. This complements our result (1), and shows that the running time of the SipserGácsLautemann simulation is optimal, up to a log t factor, for relativizing techniques. (All the results in (1) relativize.) This result is obtained as a corollary from a new circuit lower bound for approximate majority: poly(n)size depth3 circuits for approximate majority have bottom fanin Ω(log n). 3. We prove that solving QSAT 3 ∈ Σ3Time(n · poly log n) requires time n 1+Ω(1) on probabilistic Turing machines using space n.9, with random access to input and work tapes, and twoway sequential access to the randombit tape. This is the first lower bound of the form t = n 1+Ω(1) on a model with random access to the input and twoway access to the random bits. 4. We prove that solving QSAT 3 ∈ Σ3Time(n · poly log n) requires time n 1+Ω(1) on Turing machines with an input tape and a sequential work tape that is initialized with random bits. This is the first lower bound on a probabilistic extension of the offline Turing machine model with one work tape.
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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Cited by 2 (1 self)
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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.
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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Cited by 1 (0 self)
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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