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27
On the complexity of numerical analysis
 IN PROC. 21ST ANN. IEEE CONF. ON COMPUTATIONAL COMPLEXITY (CCC ’06
, 2006
"... We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation ..."
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Cited by 48 (7 self)
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We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model ” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a divisionfree straightline program producing an integer N, decide whether N> 0. • In the BlumShubSmale model, polynomial time computation over the reals (on discrete inputs) is polynomialtime equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. • The Generic Task of Numerical Computation is also polynomialtime equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.
On the Hardness of Graph Isomorphism
 SIAM J. COMPUT
"... We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 manyone reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the stro ..."
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Cited by 32 (1 self)
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We show that the graph isomorphism problem is hard under DLOGTIME uniform AC0 manyone reductions for the complexity classes NL, PL (probabilistic logarithmic space) for every logarithmic space modular class ModkL and for the class DET of problems NC¹ reducible to the determinant. These are the strongest known hardness results for the graph isomorphism problem and imply a randomized logarithmic space reduction from the perfect matching problem to graph isomorphism. We also investigate hardness results for the graph automorphism problem.
The complexity of membership problems for circuits over sets of natural numbers
, 2007
"... The problem of testing membership in the subset of the natural numbers produced at the output gate of a {∪, ∩, − , +, ×} combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case {∪, ∩, +, ×} is shown NEXPTIMEcomplete, the cas ..."
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Cited by 16 (0 self)
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The problem of testing membership in the subset of the natural numbers produced at the output gate of a {∪, ∩, − , +, ×} combinational circuit is shown to capture a wide range of complexity classes. Although the general problem remains open, the case {∪, ∩, +, ×} is shown NEXPTIMEcomplete, the cases {∪, ∩, − , ×}, {∪, ∩, ×}, {∪, ∩, +} are shown PSPACEcomplete, the case {∪, +} is shown NPcomplete, the case {∩, +} is shown C=Lcomplete, and several other cases are resolved. Interesting auxiliary problems are used, such as testing nonemptyness for unionintersectionconcatenation circuits, and expressing each integer, drawn from a set given as input, as powers of relatively prime integers of one’s choosing. Our results extend in nontrivial ways past work by
Amplifying lower bounds by means of selfreducibility
 In IEEE Conference on Computational Complexity
, 2008
"... We observe that many important computational problems in NC 1 share a simple selfreducibility property. We then show that, for any problem A having this selfreducibility property, A has polynomial size TC 0 circuits if and only if it has TC 0 circuits of size n 1+ɛ for every ɛ>0 (counting the numb ..."
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Cited by 13 (4 self)
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We observe that many important computational problems in NC 1 share a simple selfreducibility property. We then show that, for any problem A having this selfreducibility property, A has polynomial size TC 0 circuits if and only if it has TC 0 circuits of size n 1+ɛ for every ɛ>0 (counting the number of wires in a circuit as the size of the circuit). As an example of what this observation yields, consider the Boolean Formula Evaluation problem (BFE), which is complete for NC 1 and has the selfreducibility property. It follows from a lower bound of Impagliazzo, Paturi, and Saks, that BFE requires depth d TC 0 circuits of size n 1+ɛd. If one were able to improve this lower bound to show that there is some constant ɛ>0 such that every TC 0 circuit family recognizing BFE has size n 1+ɛ, then it would follow that TC 0 ̸ = NC 1. We show that proving lower bounds of the form n 1+ɛ is not ruled out by the Natural Proof framework of Razborov and Rudich and hence there is currently no known barrier for separating classes such as ACC 0,TC 0 and NC 1 via existing “natural ” approaches to proving circuit lower bounds. We also show that problems with small uniform constantdepth circuits have algorithms that simultaneously have small space and time bounds. We then make use of known timespace tradeoff lower bounds to show that SAT requires uniform depth d TC 0 and AC 0 [6] circuits of size n 1+c for some constant c depending on d. 1
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 12 (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
Constantdepth circuits for arithmetic in finite fields of characteristic two
 In Proceedings of the 23rd International Symposium on Theoretical Aspects of Computer Science (STACS), Lecture Notes in Computer Science
, 2006
"... We study the complexity of arithmetic in finite fields of characteristic two, F2n. We concentrate on the following two problems: • Iterated Multiplication: Given α1, α2,...,αt ∈ F2 n, compute α1 · α2 · · ·αt ∈ F2 n. • Exponentiation: Given α ∈ F2 n and a tbit integer k, compute αk ∈ F2 n. ..."
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Cited by 11 (6 self)
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We study the complexity of arithmetic in finite fields of characteristic two, F2n. We concentrate on the following two problems: • Iterated Multiplication: Given α1, α2,...,αt ∈ F2 n, compute α1 · α2 · · ·αt ∈ F2 n. • Exponentiation: Given α ∈ F2 n and a tbit integer k, compute αk ∈ F2 n.
Derandomization and Distinguishing Complexity
, 2003
"... We continue an investigation of resourcebounded Kolmogorov complexity and derandomization techniques begun in [2, 3]. ..."
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Cited by 10 (5 self)
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We continue an investigation of resourcebounded Kolmogorov complexity and derandomization techniques begun in [2, 3].
On defining integers and proving arithmetic circuit lower bounds
 Computational Complexity
"... Abstract. Let τ(n) denote the minimum number of arithmetic operations sufficient to build the integer n from the constant 1. We prove that if there are arithmetic circuits of size polynomial in n for computing the permanent of n by n matrices, then τ(n!) is polynomially bounded in log n. Under the s ..."
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Cited by 8 (0 self)
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Abstract. Let τ(n) denote the minimum number of arithmetic operations sufficient to build the integer n from the constant 1. We prove that if there are arithmetic circuits of size polynomial in n for computing the permanent of n by n matrices, then τ(n!) is polynomially bounded in log n. Under the same assumption on the permanent, we conclude that the PochhammerWilkinson polynomials ∏n k=1 (X − k) and the Taylor approximations ∑n k=0 1 k! Xk and ∑n k=1 1 k Xk of exp and log, respectively, can be computed by arithmetic circuits of size polynomial in log n (allowing divisions). This connects several so far unrelated conjectures in algebraic complexity.
The Pervasive Reach of ResourceBounded Kolmogorov Complexity in Computational Complexity Theory
"... We continue an investigation into resourcebounded Kolmogorov complexity [ABK + 06], which highlights the close connections between circuit complexity and Levin’s timebounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to ..."
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Cited by 6 (1 self)
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We continue an investigation into resourcebounded Kolmogorov complexity [ABK + 06], which highlights the close connections between circuit complexity and Levin’s timebounded Kolmogorov complexity measure Kt (and other measures with a similar flavor), and also exploits derandomization techniques to provide new insights regarding Kolmogorov complexity. The Kolmogorov measures that have been introduced have many advantages over other approaches to defining resourcebounded Kolmogorov complexity (such as much greater independence from the underlying choice of universal machine that is used to define the measure) [ABK + 06]. Here, we study the properties of other measures that arise naturally in this framework. The motivation for introducing yet more notions of resourcebounded Kolmogorov complexity are twofold: • to demonstrate that other complexity measures such as branchingprogram size and formula size can also be discussed in terms of Kolmogorov complexity, and • to demonstrate that notions such as nondeterministic Kolmogorov complexity and distinguishing complexity [BFL02] also fit well into this framework. The main theorems that we provide using this new approach to resourcebounded Kolmogorov complexity are: • A complete set (RKNt) for NEXP/poly defined in terms of strings of high Kolmogorov complexity.
Uniform Derandomization from Pathetic Lower Bounds
, 2009
"... A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e., superpolynomial, or even nearlyexponential) circuit size lower bounds for certain problems. In contrast to what is needed ..."
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Cited by 2 (2 self)
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A recurring theme in the literature on derandomization is that probabilistic algorithms can be simulated quickly by deterministic algorithms, if one can obtain impressive (i.e., superpolynomial, or even nearlyexponential) circuit size lower bounds for certain problems. In contrast to what is needed for derandomization, existing lower bounds seem rather pathetic (linearsize lower bounds for general circuits [IM02], nearly cubic lower bounds for formula size [H˚as98], nearly n log log n size lower bounds for branching programs [BSSV03], n 1+cd for depth d threshold circuits [IPS97]). Here, we present two instances where “pathetic ” lower bounds of the form n 1+ɛ would suffice to derandomize interesting classes of probabilistic algorithms. We show: • If the word problem over S5 requires constantdepth threshold circuits of size n1+ɛ for some ɛ> 0, then any language accepted by uniform polynomialsize probabilistic threshold circuits is accepted by a uniform family of deterministic constantdepth threshold circuits of subexponential size. • If there are no constantdepth arithmetic circuits of size n1+ɛ for the problem of multiplying a sequence of n 3by3 matrices, then for every constant d, blackbox identity testing for depthd arithmetic circuits with bounded individual degree can be performed by a uniform family of deterministic constantdepth AC0 circuits of subexponential size.