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29
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
List Decoding of AlgebraicGeometric Codes
 IEEE Trans. on Information Theory
, 1999
"... We generalize Sudan's results for ReedSolomon codes to the class of algebraicgeometric codes, designing algorithms for list decoding of algebraic geometric codes which can decode beyond the conventional errorcorrection bound (d\Gamma1)=2, d being the minimumdistance of the code. Our main algorith ..."
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Cited by 41 (3 self)
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We generalize Sudan's results for ReedSolomon codes to the class of algebraicgeometric codes, designing algorithms for list decoding of algebraic geometric codes which can decode beyond the conventional errorcorrection bound (d\Gamma1)=2, d being the minimumdistance of the code. Our main algorithm is based on an interpolation scheme and factorization of polynomials over algebraic function fields. For the latter problem we design a polynomialtime algorithm and show that the resulting overall listdecoding algorithm runs in polynomial time under some mild conditions. Several examples are included.
Factorization of Polynomials Given by StraightLine Programs
 Randomness and Computation
, 1989
"... An algorithm is developed for the factorization of a multivariate polynomial represented by traightline program into its irreducible factors. The algorithm is in random polynomialtime as a function in the input size, total degree, and binary coefficient length for the usual coefficient fields and ..."
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Cited by 29 (9 self)
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An algorithm is developed for the factorization of a multivariate polynomial represented by traightline program into its irreducible factors. The algorithm is in random polynomialtime as a function in the input size, total degree, and binary coefficient length for the usual coefficient fields and outputs a straightline program, which with controllably high probability correctly determines the irreducible factors. It also returns the probably correct multiplicities of each distinct factor. If th oefficient field has finite characteristic p and p divides the multiplicities of some irreducible factors our algorithm constructs straightline programs for the appropriate pth powers of such factors. Also a probabilistic algorithm is presented that allows to convert a polynomial given by a straightline program into its sparse representation. This conversion algorithm is in randompolynomial time in the previously cited parameters and in an upper bound for the number of nonzero...
A polynomialtime complexity bound for the computation of the singular part of a Puiseux expansion of an algebraic function
 MR 2000j:14098
"... Dedicated to Wolfgang Schmidt on the occasion of his sixtieth birthday. Abstract. In this paper we present a refined version of the Newton polygon process to compute the Puiseux expansions of an algebraic function defined over the rational function field. We determine an upper bound for the bitcompl ..."
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Cited by 14 (1 self)
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Dedicated to Wolfgang Schmidt on the occasion of his sixtieth birthday. Abstract. In this paper we present a refined version of the Newton polygon process to compute the Puiseux expansions of an algebraic function defined over the rational function field. We determine an upper bound for the bitcomplexity of computing the singular part of a Puiseux expansion by this algorithm, and use a recent quantitative version of Eisenstein’s theorem on power series expansions of algebraic functions to show that this computational complexity is polynomial in the degrees and the logarithm of the height of the polynomial defining the algebraic function. 1.
Decoding AlgebraicGeometric Codes Beyond the ErrorCorrection Bound
, 1998
"... Generalizing the highnoise decoding methods of [1, 19] to the class of algebraicgeometric codes, we design the first polynomialtime algorithms to decode algebraicgeometric codes significantly beyond the conventional errorcorrection bound. Applying our results to codes obtained from curves with m ..."
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Cited by 13 (4 self)
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Generalizing the highnoise decoding methods of [1, 19] to the class of algebraicgeometric codes, we design the first polynomialtime algorithms to decode algebraicgeometric codes significantly beyond the conventional errorcorrection bound. Applying our results to codes obtained from curves with many rational points, we construct arbitrarily long, constantrate linear codes over a fixed field F q such that a codeword is efficiently, nonuniquely reconstructible after a majority of its letters have been arbitrarily corrupted. We also construct codes such that a codeword is uniquely and efficiently reconstructible after a majority of its letters have been corrupted by noise which is random in a specified sense. We summarize our results in terms of bounds on asymptotic parameters, giving a new characterization of decoding beyond the errorcorrection bound. 1 Introduction Errorcorrecting codes, originally designed to accommodate reliable transmission of information through unreliable ...
Effective Hilbert Irreducibility
, 1985
"... n this paper we prove by entirely elementary means a very effective version of the Hilbert Irreducibility  n Theorem. We then apply our theorem to construct a probabilistic irreducibility test for sparse multivariate poly omials over arbitrary perfect fields. For the usual coefficient fields the te ..."
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Cited by 13 (6 self)
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n this paper we prove by entirely elementary means a very effective version of the Hilbert Irreducibility  n Theorem. We then apply our theorem to construct a probabilistic irreducibility test for sparse multivariate poly omials over arbitrary perfect fields. For the usual coefficient fields the test runs in polynomial time in the input K size. eywords. Hilbert Irreducibility Theorem, Probabilistic Algorithms, Polynomial Factorization, Sparse Polynomials. 1. Introduction s i The question whether a polynomial with coefficients in a unique factorization domain i rreducible poses an old problem. Recently, several new algorithms for univariate and mul  w tivariate factorization over various coefficient domains have been proposed within the frame ork of polynomial time complexity, see e.g. Berlekamp (1970), Lenstra et al. (1982), Kaltoj fen (1985a), Chistov and Grigoryev (1982), Landau (1985). All algorithms in the references ust given are polynomial in l (n +1) , where l is the num...
Simplification of Nested Radicals
, 1993
"... Radical simplification is an important part of symbolic computation systems. Until now no algorithms were known for the general denesting problem. If the base field contains all roots of unity, then we give necessary and sufficient conditions for a denesting, and our algorithm computes a denesti ..."
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Cited by 12 (0 self)
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Radical simplification is an important part of symbolic computation systems. Until now no algorithms were known for the general denesting problem. If the base field contains all roots of unity, then we give necessary and sufficient conditions for a denesting, and our algorithm computes a denesting of ff when it exists. If the base field does not contain all roots of unity, then we show how to compute a denesting that is within depth one of optimal over the base field adjoin a single root of unity. Throughout our paper, we choose to represent a primitive l th root of unity by its symbol i l , rather than as a nested radical. The algorithms require computing the splitting field of the minimal polynomial of ff over k, and have exponential running time. In his magic way, Ramanujan observed a number of striking relationships between certain nested radicals: 3 q 3 p 2 \Gamma 1 = 3 q 1=9 \Gamma 3 q 2=9 + 3 q 4=9 (1) q 3 p 5 \Gamma 3 p 4 = 1=3( 3 p 2 + 3 p 20 ...
Multiplicative Equations Over Commuting Matrices
, 1995
"... We consider the solvability of the equation k Y i=1 A i x i = B and generalizations, where the A i and B are given commuting matrices over an algebraic number field F . In the semigroup membership problem, the variables x i are constrained to be nonnegative integers. While this problem is NPco ..."
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Cited by 11 (4 self)
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We consider the solvability of the equation k Y i=1 A i x i = B and generalizations, where the A i and B are given commuting matrices over an algebraic number field F . In the semigroup membership problem, the variables x i are constrained to be nonnegative integers. While this problem is NPcomplete for variable k, we give a polynomial time algorithm if k is fixed. In the group membership problem, the matrices are assumed to be invertible, and the variables x i may take on negative values. In this case we give a polynomial time algorithm for variable k and give an explicit description of the set of all solutions (as an affine lattice). The results generalize recent work of Cai, Lipton, and Zalcstein [CLZ] where the case k = 2 is solved using Jordan Normal Forms (JNF). We achieve greater clarity, simplicity, and generality by eliminating the use of JNF's and referring to elementary concepts of the structure theory of algebras instead (notably, the radical and the local decomposit...