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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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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
Progress on Polynomial Identity Testing
"... Polynomial identity testing (PIT) is the problem of checking whether a given arithmetic circuit is the zero circuit. PIT ranks as one of the most important open problems in the intersection of algebra and computational complexity. In the last few years, there has been an impressive progress on this ..."
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Polynomial identity testing (PIT) is the problem of checking whether a given arithmetic circuit is the zero circuit. PIT ranks as one of the most important open problems in the intersection of algebra and computational complexity. In the last few years, there has been an impressive progress on this problem but a complete solution might take a while. In this article we give a soft survey exhibiting the ideas that have been useful. 1
A Hitting Set Construction, with Applications to Arithmetic Circuit Lower Bounds
, 2009
"... Abstract. A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic blackbox identity testing algorithm for univariate polynomials of the form Pt j=0 cjXα j β (a + bX) j. From our algorithm we derive Q an exponential l ..."
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Abstract. A polynomial identity testing algorithm must determine whether a given input polynomial is identically equal to 0. We give a deterministic blackbox identity testing algorithm for univariate polynomials of the form Pt j=0 cjXα j β (a + bX) j. From our algorithm we derive Q an exponential lower bound for representations of polynomials such as n 2 i=1 (Xi − 1) under this form. It has been conjectured that these polynomials are hard to compute by general arithmetic circuits. Our result shows that the “hardness from derandomization” approach to lower bounds is feasible for a restricted class of arithmetic circuits. The proof is based on techniques from algebraic number theory, and more precisely on properties of the height function of algebraic numbers. 1
Randomness Efficient Testing of Sparse Black Box Identities of Unbounded Degree over the Reals
"... We construct a hitting set generator for sparse multivariate polynomials over the reals. The seed length of our generator is O(log2(mn/)) where m is the number of monomials, n is number of variables, and 1 − is the hitting probability. The generator can be evaluated in time polynomial in logm, n, ..."
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We construct a hitting set generator for sparse multivariate polynomials over the reals. The seed length of our generator is O(log2(mn/)) where m is the number of monomials, n is number of variables, and 1 − is the hitting probability. The generator can be evaluated in time polynomial in logm, n, and log 1/. This is the first hitting set generator whose seed length is independent of the degree of the polynomial. The seed length of the best generator so far by Klivans and Spielman [16] depends logarithmically on the degree. From this, we get a randomized algorithm for testing sparse black box polynomial identities over the reals using O(log2(mn/)) random bits with running time polynomial in logm, n, and log 1 . We also design a deterministic test with running time Õ(m3n3). Here, the Õnotation suppresses polylogarithmic factors. The previously best deterministic test by Lipton and Vishnoi [18] has a running time that depends polynomially on log δ, where δ is the degree of the black box polynomial.
Interpolation of ShiftedLacunary Polynomials
, 2010
"... Given a “black box” function to evaluate an unknown rational polynomial f ∈ Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t ∈ Z>0, the shift α ∈ Q, the exponents 0 ≤ e1 ..."
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Given a “black box” function to evaluate an unknown rational polynomial f ∈ Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t ∈ Z>0, the shift α ∈ Q, the exponents 0 ≤ e1 <e2 < ·· · <et, and the coefficients c1,...,ct ∈ Q \{0} such that f(x) =c1(x − α) e1 + c2(x − α) e2 + ···+ ct(x − α) et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size, which may be logarithmic in the degree of f. Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information.