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103
Lower bounds in a parallel model without bit operations
 TO APPEAR IN THE SIAM JOURNAL ON COMPUTING
, 1997
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A Lower Bound for the Size of Syntactically Multilinear Arithmetic Circuits
 Proceedings of the 48th FOCS: 438–448
, 2007
"... We construct an explicit polynomial f(x1,...,xn), with coefficients in {0, 1}, such that the size of any syntactically multilinear arithmetic circuit computing f is at least Ω(n 4/3 / log 2 n). The lower bound holds over any field. ..."
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Cited by 14 (10 self)
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We construct an explicit polynomial f(x1,...,xn), with coefficients in {0, 1}, such that the size of any syntactically multilinear arithmetic circuit computing f is at least Ω(n 4/3 / log 2 n). The lower bound holds over any field.
Computing the sign or the value of the determinant of an integer matrix, a complexity survey
 JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
, 2004
"... Computation of the sign of the determ9vSof amWDz[ and the determBvSitself is a challenge for both numhvB5q and exact mactv;W We survey the comWzqDvS of existingmistin to solve these problem when the input is an nnm;9q; A with integer entries. We study the bitcomvSW5[5Wv of the algorithm asymithmW5[ ..."
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Cited by 13 (3 self)
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Computation of the sign of the determ9vSof amWDz[ and the determBvSitself is a challenge for both numhvB5q and exact mactv;W We survey the comWzqDvS of existingmistin to solve these problem when the input is an nnm;9q; A with integer entries. We study the bitcomvSW5[5Wv of the algorithm asymithmW5[5 n and thenorm of A. Existing approaches rely onnumBDWzv approximW[ comoximW[z5 on exactcomvB[;5vSW or on both types of arithmvSW incom9zqvSWqD c 2003 Elsevier B.V. All rights reserved. Keywords: Determ9v9vm Bitcom9vmv Integer mteger Approxim59 comoxim599 Exact comv55DvS Random5D algorithm 1. I517251716 Com;vm9 the sign or the value of thedetermDvS; nmBqq A is a classicalproblem Numblem mmble are usually focused oncomBW9v the sign via an accurateapproxim;B99 of the determvS;;5 Amer the applications areimvWD;qW problem ofcom9qWvS;;5[ geom9qW that can be reduced to the determ5vS; question; the readerma refer to [11,12,9,10,46,45] and to the bibliography therein. InsymDW;B comW;BvS55 theproblem ofcomDzWv the exact value of the ThismisvqB; is based on work supported in part by the National Science Foundation under grants Nrs. DMS9977392, CCR9988177, and CCR0113121 (Kaltofen) and by the Centre National de la Recherche Scienti#que, Actions Incitatives No. 5929 et STIC LINBOX 2001 (Villard).
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.
From a zoo to a zoology: Towards a general theory of graph polynomials
 Theory of Computing Systems
, 2007
"... Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We in ..."
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Cited by 12 (5 self)
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Abstract. We outline a general theory of graph polynomials which covers all the examples we found in the vast literature, in particular, the chromatic polynomial, various generalizations of the Tutte polynomial, matching polynomials, interlace polynomials, and the cover polynomial of digraphs. We introduce two classes of (hyper)graph polynomials definable in second order logic, and outline a research program for their classification in terms of definability and complexity considerations, and various notions of reducibilities. 1
Circuit lower bounds for MerlinArthur classes
 In Proc. ACM STOC
, 2007
"... We show that for each k> 0, MA/1 (MA with 1 bit of advice) doesn’t have circuits of size nk. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM and ZPP NP We extend our main result in several ways. For each k, we give an explicit language in (MA ∩ ..."
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Cited by 12 (1 self)
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We show that for each k> 0, MA/1 (MA with 1 bit of advice) doesn’t have circuits of size nk. This implies the first superlinear circuit lower bounds for the promise versions of the classes MA, AM and ZPP NP We extend our main result in several ways. For each k, we give an explicit language in (MA ∩ coMA)/1 which doesn’t have circuits of size nk. We also adapt our lower bound to the averagecase setting, i.e., we show that MA/1 cannot be solved on more than 1/2 + 1/nk fraction of inputs of length n by circuits of size nk. Furthermore, we prove that MA does not have arithmetic circuits of size nk for any k. As a corollary to our main result, we obtain that derandomization of MA with O(1) advice implies the existence of pseudorandom generators computable using O(1) bits of advice. 1
Hardness as randomness: A survey of universal derandomization
 in Proceedings of the International Congress of Mathematicians
, 2002
"... We survey recent developments in the study of probabilistic complexity classes. While the evidence seems to support the conjecture that probabilism can be deterministically simulated with relatively low overhead, i.e., that P = BPP, it also indicates that this may be a difficult question to resolve. ..."
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Cited by 11 (5 self)
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We survey recent developments in the study of probabilistic complexity classes. While the evidence seems to support the conjecture that probabilism can be deterministically simulated with relatively low overhead, i.e., that P = BPP, it also indicates that this may be a difficult question to resolve. In fact, proving that probalistic algorithms have nontrivial deterministic simulations is basically equivalent to proving circuit lower bounds, either in the algebraic or Boolean models.
Efficient Zeroknowledge Authentication Based on a Linear Algebra Problem MiniRank
 ADVANCES IN CRYPTOLOGY – ASIACRYPT 2001, VOLUME 2248 OF LECTURE NOTES IN COMPUTER SCIENCE
, 2001
"... A Zeroknowledge protocol provides provably secure entity authentication based on a hard computational problem. Among many schemes proposed since 1984, the most practical rely on factoring and discrete log, but still they are practical schemes based on NPhard problems. Among them, ..."
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Cited by 11 (0 self)
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A Zeroknowledge protocol provides provably secure entity authentication based on a hard computational problem. Among many schemes proposed since 1984, the most practical rely on factoring and discrete log, but still they are practical schemes based on NPhard problems. Among them,
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
On the algebraic complexity of some families of coloured Tutte polynomials
 Advances in Applied Mathematics
, 2004
"... Abstract. We investigate the coloured Tutte polynomial in Valiant’s algebraic framework of NPcompleteness. Generalising the well known relationship between the Tutte polynomial and the partition function from the Ising model, we establish a reduction from the permanent to the coloured Tutte polynom ..."
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Cited by 10 (4 self)
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Abstract. We investigate the coloured Tutte polynomial in Valiant’s algebraic framework of NPcompleteness. Generalising the well known relationship between the Tutte polynomial and the partition function from the Ising model, we establish a reduction from the permanent to the coloured Tutte polynomial, thus showing that its evaluation is a VNP−complete problem.