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28
Nonuniform ACC circuit lower bounds
, 2010
"... The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipoly ..."
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The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynomials and other less natural functions. • ENP, the class of languages recognized in 2O(n) time with an NP oracle, doesn’t have nonuniform ACC circuits of 2no(1) size. The lower bound gives an exponential sizedepth tradeoff: for every d there is a δ> 0 such that ENP doesn’t have depthd ACC circuits of size 2nδ. Previously, it was not known whether EXP NP had depth3 polynomial size circuits made out of only MOD6 gates. The highlevel strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail the above lower bounds. The algorithm combines known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a subtle strengthening of the author’s prior work [STOC’10]. Supported by the Josef Raviv Memorial Fellowship.
Improving Exhaustive Search Implies Superpolynomial Lower Bounds
, 2009
"... The P vs NP problem arose from the question of whether exhaustive search is necessary for problems with short verifiable solutions. We do not know if even a slight algorithmic improvement over exhaustive search is universally possible for all NP problems, and to date no major consequences have been ..."
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Cited by 37 (7 self)
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The P vs NP problem arose from the question of whether exhaustive search is necessary for problems with short verifiable solutions. We do not know if even a slight algorithmic improvement over exhaustive search is universally possible for all NP problems, and to date no major consequences have been derived from the assumption that an improvement exists. We show that there are natural NP and BPP problems for which minor algorithmic improvements over the trivial deterministic simulation already entail lower bounds such as NEXP ̸ ⊆ P/poly and LOGSPACE ̸ = NP. These results are especially interesting given that similar improvements have been found for many other hard problems. Optimistically, one might hope our results suggest a new path to lower bounds; pessimistically, they show that carrying out the seemingly modest program of finding slightly better algorithms for all search problems may be extremely difficult (if not impossible). We also prove unconditional superpolynomial timespace lower bounds for improving on exhaustive search: there is a problem verifiable with k(n) length witnesses in O(n a) time (for some a and some function k(n) ≤ n) that cannot be solved in k(n) c n a+o(1) time and k(n) c n o(1) space, for every c ≥ 1. While such problems can always be solved by exhaustive search in O(2 k(n) n a) time and O(k(n) + n a) space, we can prove a superpolynomial lower bound in the parameter k(n) when space usage is restricted.
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 22 (8 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 15 (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.
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 15 (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.
Inductive TimeSpace Lower Bounds for SAT and Related Problems
 Computational Complexity
, 2005
"... Abstract. We improve upon indirect diagonalization arguments for lower bounds on explicit problems within the polynomial hierarchy. Our contributions are summarized as follows. 1. We present a technique that uniformly improves upon most known nonlinear time lower bounds for nondeterminism andalterna ..."
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Cited by 14 (4 self)
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Abstract. We improve upon indirect diagonalization arguments for lower bounds on explicit problems within the polynomial hierarchy. Our contributions are summarized as follows. 1. We present a technique that uniformly improves upon most known nonlinear time lower bounds for nondeterminism andalternating computation, on both subpolynomial (n o(1) ) space RAMs and sequential onetape machines with random access to the input. We obtain improved lower bounds for Boolean satisfiability (SAT), as well as all NPcomplete problems that have efficient reductions from SAT, and ΣkSAT, for constant k ≥ 2. For example, SAT cannot be solved by random access machines using n √ 3 time and subpolynomial space. 2. We show how indirect diagonalization leads to timespace lower bounds for computation with bounded nondeterminism. For both the random access and multitape Turing machine models, we prove that for all k ≥ 1, there is a constant ck> 1 such that linear time with n 1/k nondeterministic bits is not contained in deterministic n ck time with subpolynomial space. This is used to prove that satisfiability of Boolean circuits with n inputs and n k size cannot be solved by deterministic multitape Turing machines running in n k·ck time and subpolynomial space.
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 11 (5 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 7 (0 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
Hardness of approximation for quantum problems
 In Proceedings of 39th International Colloquium on Automata, Languages and Programming (ICALP 2012
, 2012
"... Abstract The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that the ..."
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Abstract The polynomial hierarchy plays a central role in classical complexity theory. Here, we define a quantum generalization of the polynomial hierarchy, and initiate its study. We show that not only are there natural complete problems for the second level of this quantum hierarchy, but that these problems are in fact strongly hard to approximate. Our results thus yield the first known hardness of approximation results for a quantum complexity class. Our approach is based on the use of dispersers, and is inspired by the classical results of Umans regarding hardness of approximation for the second level of the classical polynomial hierarchy [Umans, FOCS 1999]. Over the last decades, the Polynomial Hierarchy (PH) [MS72], a natural generalization of the class NP, has been the focus of much study in classical computational complexity. Of particular interest is the second level of PH, denoted Σ p 2 . Here, we say a problem is in Σ p 2 if it has an efficient verifier with the property that for any YES instance x ∈ {0, 1} * of the problem, there exists a polynomial length proof y such that for all polynomial length proofs z, the verifier accepts x, y and z. Note the alternation from an existential quantifier over y to a forall quantifier over z is crucial here keeping only the existential quantifier reduces us to NP. It turns out that the use of such alternating quantifiers makes Σ p 2 a powerful class which is believed to be beyond NP. For example, many natural and important problems are known to be in Σ p 2 but not in NP. Such problems range from "does the optimal assignment to a 3SAT instance satisfy exactly k clauses?" to extremely practically relevant problems related to circuit minimization, such as "given a boolean formula C in Disjunctive Normal Form (DNF), what is the smallest DNF formula C equivalent to C?". Further, the study of Σ p 2 has led to a host of other fundamental theoretical results, such as the KarpLipton theorem, which states that NP ⊆ P /poly unless PH collapses to Σ p 2 . Σ p 2 has even been used, for example, to prove that SAT cannot be solved simultaneously in linear time and logarithmic space [For00, FLvMV05]. For these reasons, Σ p 2 and more generally PH have occupied a central role in classical complexity theoretic research. Moving to the quantum setting, the study of quantum proof systems and a natural quantum generalization of NP, the class Quantum Merlin Arthur (QMA) [KSV02], has been a very active area of research over the last decade. Roughly, a problem is in QMA if for any YES instance of the problem, there exists a polynomial size quantum proof convincing a quantum verifier of this fact with high probability. With the notion of quantum proofs in mind, we thus ask the natural question: Can a quantum generalization of Σ p 2 be defined, and what types of problems might it contain and characterize? Perhaps surprisingly, to date there are almost no known results in this direction. Our results: In this work, we introduce a quantum generalization of Σ p 2 , which we call cqΣ p 2 , and initiate its study. In particular, we study cqΣ p 2 completeness and moreover cqΣ p 2 hardness of approximation for two new problems we define. In order to state our first main result, Theorem 5 (our second similar result for the problem QIRR will appear in a technical version of this abstract), we begin by introducing the requisite definitions. First, the class cqΣ p 2 is informally defined as: if it has an efficient quantum verifier satisfying the following property for any input x ∈ {0, 1} * : • If x is a YES instance of Π, then there exists a polynomial length classical proof y such that for all polynomial length quantum proofs z , the verifier accepts x, y and z with high probability.
An Improved TimeSpace Lower Bounds for Tautologies
"... We show that for all reals c and d such that c 2 d < 4 there exists a positive real e such that tautologies of length n 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 d & ..."
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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 of length n 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 d < 3√ 4 there exists a positive e such that tautologies cannot be decided by a nondeterministic algorithm that runs in time n d and space n e.