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36
Proving lower bounds via pseudorandom generators
 FSTTCS 2005: Foundations of Software Technology and Theoretical Computer Science, 25th International Conference, Hyderabad, India, December 1518, 2005, Proceedings, volume 3821 of Lecture
, 2005
"... Abstract. In this paper, we formalize two stepwise approaches, based on pseudorandom generators, for proving P � = NP and its arithmetic analog: Permanent requires superpolynomial sized arithmetic circuits. 1 ..."
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Cited by 31 (1 self)
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Abstract. In this paper, we formalize two stepwise approaches, based on pseudorandom generators, for proving P � = NP and its arithmetic analog: Permanent requires superpolynomial sized arithmetic circuits. 1
Holographic Algorithms: From Art to Science
 Electronic Colloquium on Computational Complexity Report
, 2007
"... We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to ..."
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Cited by 19 (10 self)
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We develop the theory of holographic algorithms. We give characterizations of algebraic varieties of realizable symmetric generators and recognizers on the basis manifold, and a polynomial time decision algorithm for the simultaneous realizability problem. Using the general machinery we are able to give unexpected holographic algorithms for some counting problems, modulo certain Mersenne type integers. These counting problems are #Pcomplete without the moduli. Going beyond symmetric signatures, we define dadmissibility and drealizability for general signatures, and give a characterization of 2admissibility and some general constructions of admissible and realizable families. 1
AN OVERVIEW OF MATHEMATICAL ISSUES ARISING IN THE GEOMETRIC COMPLEXITY THEORY APPROACH TO VP != VNP
"... We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant’s algebraic analog of the P ̸ = NP conjecture. We also describe the precise separation of complexity classes that t ..."
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Cited by 15 (6 self)
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We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant’s algebraic analog of the P ̸ = NP conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate.
A quadratic bound for the determinant and permanent problem
 International Mathematics Research Notices
, 2004
"... The size of an arithmetical formula is the number of symbols (+, ×) which it contains. ..."
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Cited by 13 (0 self)
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The size of an arithmetical formula is the number of symbols (+, ×) which it contains.
Geometric Complexity Theory VI: the flip via saturated and positive integer programming in representation theory and algebraic geometry
, 2009
"... ..."
Arithmetic Circuits: a survey of recent results and open questions
"... A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last fi ..."
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Cited by 11 (3 self)
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A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions. The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we
Is P versus NP formally independent
 Bulletin of the European Association for Theoretical Computer Science
, 2003
"... I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy! ..."
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Cited by 8 (0 self)
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I have moved back to the University of Chicago and so has the web page for this column. See above for new URL and contact informaion. This issue Scott Aaronson writes quite an interesting (and opinionated) column on whether the P = NP question is independent of the usual axiom systems. Enjoy!
Geometric complexity theory III: on deciding positivity of LittlewoodRichardson coefficients
, 2005
"... We point out that the remarkable Knutson and Tao Saturation Theorem [9] and polynomial time algorithms for linear programming [14] have together an important, immediate consequence in geometric complexity theory [15, 16]: The problem of deciding positivity of LittlewoodRichardson coefficients belon ..."
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Cited by 8 (6 self)
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We point out that the remarkable Knutson and Tao Saturation Theorem [9] and polynomial time algorithms for linear programming [14] have together an important, immediate consequence in geometric complexity theory [15, 16]: The problem of deciding positivity of LittlewoodRichardson coefficients belongs to P; cf.[10]. Specifically, for GLn(C), positivity of a LittlewoodRichardson coefficient cα,β,γ can be decided in time that is polynomial in n and the bit lengths of the specifications of the partitions α, β and γ. Furthermore, the algorithm is strongly polynomial in the sense of [14]. The main goal of this article is to explain the significance of this result in the context of geometric complexity theory. Furthermore, it is also conjectured that an analogous result holds for arbitrary symmetrizable KacMoody algebras.