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19
Power from Random Strings
 IN PROCEEDINGS OF THE 43RD IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE
, 2002
"... We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and nonuniform reductions. These sets are provably not complete under the usual manyone reductions. Let ..."
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Cited by 36 (15 self)
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We show that sets consisting of strings of high Kolmogorov complexity provide examples of sets that are complete for several complexity classes under probabilistic and nonuniform reductions. These sets are provably not complete under the usual manyone reductions. Let
Locally decodable codes with 2 queries and polynomial identity testing for depth 3 circuits
 SIAM J. COMPUT
, 2007
"... In this work we study two, seemingly unrelated, notions. Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given ..."
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Cited by 26 (7 self)
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In this work we study two, seemingly unrelated, notions. Locally decodable codes (LDCs) are codes that allow the recovery of each message bit from a constant number of entries of the codeword. Polynomial identity testing (PIT) is one of the fundamental problems of algebraic complexity: we are given a circuit computing a multivariate polynomial and we have to determine whether the polynomial is identically zero. We improve known results on LDCs and on polynomial identity testing and show a relation between the two notions. In particular we obtain the following results: (1) We show that if E: F n ↦ → F m is a linear LDC with two queries, then m = exp(Ω(n)). Previously this was known only for fields of size ≪ 2 n [O. Goldreich et al., Comput. Complexity, 15 (2006), pp. 263–296]. (2) We show that from every depth 3 arithmetic circuit (ΣΠΣ circuit), C, with a bounded (constant) top fanin that computes the zero polynomial, one can construct an LDC. More formally, assume that C is minimal (no subset of the multiplication gates sums to zero) and simple (no linear function appears in all the multiplication gates). Denote by d the degree of the polynomial computed by C and by r the rank of the linear functions appearing in C. Then we can construct a linear LDC with two queries that encodes messages of length r/polylog(d) by codewords of length O(d). (3) We prove a structural theorem for ΣΠΣ circuits, with a bounded top fanin, that
Deterministic algorithms for rank aggregation and other ranking and clustering problems
 In In Proceedings of the Fifth International Workshop on Approximation and Online Algorithms
, 2007
"... Abstract. We consider ranking and clustering problems related to the aggregation of inconsistent information. Ailon, Charikar, and Newman [1] proposed randomized constant factor approximation algorithms for these problems. Together with Hegde and Jain, we recently proposed deterministic versions of ..."
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Cited by 23 (2 self)
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Abstract. We consider ranking and clustering problems related to the aggregation of inconsistent information. Ailon, Charikar, and Newman [1] proposed randomized constant factor approximation algorithms for these problems. Together with Hegde and Jain, we recently proposed deterministic versions of some of these randomized algorithms [2]. With one exception, these algorithms required the solution of a linear programming relaxation. In this paper, we introduce a purely combinatorial deterministic pivoting algorithm for weighted ranking problems with weights that satisfy the triangle inequality; our analysis is quite simple. We then shown how to use this algorithm to get the first deterministic combinatorial approximation algorithm for the partial rank aggregation problem with performance guarantee better than 2. In addition, we extend our approach to the linear programming based algorithms in Ailon et al. [1] and Ailon [3]. Finally, we show that constrained rank aggregation is not harder than unconstrained rank aggregation.
Polynomial identity testing for depth 3 circuits
 in Proceedings of the twentyfirst Annual IEEE Conference on Computational Complexity (CCC
, 2006
"... Abstract — We study ΣΠΣ(k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the first deterministic polynomial time blackbox identity test for ΣΠΣ(k) circuits over the field Q of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main ..."
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Cited by 23 (5 self)
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Abstract — We study ΣΠΣ(k) circuits, i.e., depth three arithmetic circuits with top fanin k. We give the first deterministic polynomial time blackbox identity test for ΣΠΣ(k) circuits over the field Q of rational numbers, thus resolving a question posed by Klivans and Spielman (STOC 2001). Our main technical result is a structural theorem for ΣΠΣ(k) circuits that compute the zero polynomial. In particular we show that if a ΣΠΣ(k) circuit C = ∑ i∈[k] Ai
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 constructing 11 oneway functions
 Electronic Colloquium on Computational Complexity (ECCC
, 1995
"... Abstract. We show how to construct lengthpreserving 11 oneway functions based on popular intractability assumptions (e.g., RSA, DLP). Such 11 functions should not be confused with (infinite) families of (finite) oneway permutations. What we want and obtain is a single (infinite) 11 oneway fun ..."
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Cited by 11 (1 self)
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Abstract. We show how to construct lengthpreserving 11 oneway functions based on popular intractability assumptions (e.g., RSA, DLP). Such 11 functions should not be confused with (infinite) families of (finite) oneway permutations. What we want and obtain is a single (infinite) 11 oneway function.
NP and Mathematics  a computational complexity perspective
 Proc. of the ICM 06
"... “P versus N P – a gift to mathematics from Computer Science” ..."
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Cited by 1 (0 self)
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“P versus N P – a gift to mathematics from Computer Science”
Computational Methods in Public Key Cryptology
, 2002
"... These notes informally review the most common methods from computational number theory that have applications in public key cryptology. ..."
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Cited by 1 (1 self)
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These notes informally review the most common methods from computational number theory that have applications in public key cryptology.
Randomized NPCompleteness for padic Rational Roots of Sparse Polynomials in One Variable
"... Relative to the sparse encoding, we show that deciding whether a univariate polynomial has a padic rational root can be done in NP for most inputs. We also prove a sharper complexity upper bound of P for polynomials with suitably generic padic Newton polygon. We thus improve the best previous comp ..."
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Relative to the sparse encoding, we show that deciding whether a univariate polynomial has a padic rational root can be done in NP for most inputs. We also prove a sharper complexity upper bound of P for polynomials with suitably generic padic Newton polygon. We thus improve the best previous complexity upper bound of EXPTIME. We also prove an unconditional complexity lower bound of NPhardness with respect to randomized reductions, for general univariate polynomials. The best previous lower bound assumed an unproved hypothesis on the distribution of primes in arithmetic progression. We also discuss how our results complement analogous results over the real numbers. Categories and Subject Descriptors
SERIES C: Computer ScienceYet Another Reduction from Graph to Ring Isomorphism Problems
, 2009
"... It has been known that the graph isomorphism problem is polynomialtime manyone reducible to the ring isomorphism problem. In fact, two different reductions have already been proposed. For those reductions, rings of certain types have been used to represent a given graph. In this paper, we give yet ..."
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It has been known that the graph isomorphism problem is polynomialtime manyone reducible to the ring isomorphism problem. In fact, two different reductions have already been proposed. For those reductions, rings of certain types have been used to represent a given graph. In this paper, we give yet another reduction, which is based on a simpler and more natural construction of a ring from a graph. By the existing reductions, one of the original graph isomorphisms can be found in each ring isomorphism obtained for the reduced ring isomorphism problem instance. On the other hand, in our new reduction, it is not clear how to get a graph isomorphism between two graphs from an obtained ring isomorphism between rings constructed from the graphs. However, we show that we can compute a graph isomorphism from an obtained ring isomorphism in polynomial time. In fact, one ring isomorphism may correspond to many graph isomorphisms in our reduction. Our proof essentially shows a way to obtain all graph isomorphisms corresponding to one ring isomorphism. 1