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36
Probabilistic Algorithms for Geometric Elimination
 in Engineering, Communication and Computing
, 1999
"... We develop probabilistic algorithms that solve problems of geometric elimination theory using small memory resources. These algorithms are obtained by means of the adaptation of a general transformation due to A. Borodin which converts uniform boolean circuit depth into sequential (Turing machine) s ..."
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Cited by 12 (5 self)
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We develop probabilistic algorithms that solve problems of geometric elimination theory using small memory resources. These algorithms are obtained by means of the adaptation of a general transformation due to A. Borodin which converts uniform boolean circuit depth into sequential (Turing machine) space. The boolean circuits themselves are developed using techniques based on the computation of a primitive element of a suitable zerodimensional algebra and diophantine considerations. Our algorithms improve...
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
NC algorithms for comparability graphs, interval graphs, and unique perfect matching
 Proc. 5th Conf. Found. Software Technology and Theor. Comput. Sci., volume 206 of Lect. Notes in Comput. Sci
, 1985
"... Laszlo Lovasz recently posed the following problem: \Is there an NC algorithm for testing if a given graph has a unique perfect matching?" We present suchan algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interval rep ..."
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Cited by 11 (0 self)
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Laszlo Lovasz recently posed the following problem: \Is there an NC algorithm for testing if a given graph has a unique perfect matching?" We present suchan algorithm for bipartite graphs. We also give NC algorithms for obtaining a transitive orientation of a comparability graph, and an interval representation of an interval graph. These enable us to obtain an NC algorithm for nding a maximum matching in an incomparability graph. 1
Conjunctive query answering with OWL 2 QL
 IN: PROC. OF KR. AAAI PRESS
, 2012
"... We present a novel rewriting technique for conjunctive query answering over OWL 2 QL ontologies. In general, the obtained rewritings are not necessarily correct and can be of exponential size in the length of the query. We argue, however, that in most, if not all, practical cases the rewritings are ..."
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Cited by 10 (7 self)
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We present a novel rewriting technique for conjunctive query answering over OWL 2 QL ontologies. In general, the obtained rewritings are not necessarily correct and can be of exponential size in the length of the query. We argue, however, that in most, if not all, practical cases the rewritings are correct and of polynomial size. Moreover, we prove some sufficient conditions, imposed on queries and ontologies, that guarantee correctness and succinctness. We also support our claim by experimental results.
Exponential Lower Bounds and Separation for Query Rewriting
"... We establish connections between the size of circuits and formulas computing monotone Boolean functions and the size of firstorder and nonrecursive Datalog rewritings for conjunctive queries over OWL 2 QL ontologies. We use known lower bounds and separation results from circuit complexity to prove ..."
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Cited by 8 (6 self)
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We establish connections between the size of circuits and formulas computing monotone Boolean functions and the size of firstorder and nonrecursive Datalog rewritings for conjunctive queries over OWL 2 QL ontologies. We use known lower bounds and separation results from circuit complexity to prove similar results for the size of rewritings that do not use nonsignature constants. For example, we show that, in the worst case, positive existential and nonrecursive Datalog rewritings are exponentially longer than the original queries; nonrecursive Datalog rewritings are in general exponentially more succinct than positive existential rewritings; while firstorder rewritings can be superpolynomially more succinct than positive existential rewritings.
A Compendium of Problems Complete for P
, 1991
"... This paper serves two purposes. Firstly, it is an elementary introduction to the theory of Pcompleteness  the branch of complexity theory that focuses on identifying the problems in the class P that are "hardest," in the sense that they appear to lack highly parallel solutions. That is, they ..."
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Cited by 7 (1 self)
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This paper serves two purposes. Firstly, it is an elementary introduction to the theory of Pcompleteness  the branch of complexity theory that focuses on identifying the problems in the class P that are "hardest," in the sense that they appear to lack highly parallel solutions. That is, they do not have parallel solutions using time polynomial in the logarithm of the problem size and a polynomial number of processors unless all problem in P have such solutions, or equivalently, unless P = NC . Secondly, this paper is a reference work of Pcomplete problems. We present a compilation of the known Pcomplete problems, including several unpublished or new Pcompleteness results, and many open problems. This is a preliminary version, mainly containing the problem list. The latest version of this document is available in electronic form by anonymous ftp from thorhild.cs.ualberta.ca (129.128.4.53) as either a compressed dvi file (TR9111.dvi.Z) or as a compressed postscript fi...
Sparse Hard Sets for P: Resolution of a Conjecture of Hartmanis
"... Building on a recent breakthrough by Ogihara, we resolve a conjecture made by Hartmanis in 1978 regarding the (non) existence of sparse sets complete for P under logspace manyone reductions. We show that if there exists a sparse hard set for P under logspace manyone reductions, then P = LOGSPACE. ..."
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Cited by 5 (0 self)
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Building on a recent breakthrough by Ogihara, we resolve a conjecture made by Hartmanis in 1978 regarding the (non) existence of sparse sets complete for P under logspace manyone reductions. We show that if there exists a sparse hard set for P under logspace manyone reductions, then P = LOGSPACE. We further prove that if P has a sparse hard set under manyone reductions computable in NC 1 , then P collapses to NC 1 .
On the Power of Nonlinear SecretSharing
 In Conf. on Computational Complexity
, 2001
"... A secretsharing scheme enables a dealer to distribute a secret among n parties such that only some predefined authorized sets of parties will be able to reconstruct the secret from their shares. The (monotone) collection of authorized sets is called an access structure, and is freely identified wit ..."
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Cited by 4 (4 self)
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A secretsharing scheme enables a dealer to distribute a secret among n parties such that only some predefined authorized sets of parties will be able to reconstruct the secret from their shares. The (monotone) collection of authorized sets is called an access structure, and is freely identified with its characteristic monotone function f : 1}. A family of secretsharing schemes is called efficient if the total length of the n shares is polynomial in n. Most previously known secretsharing schemes belonged to a class of linear schemes, whose complexity coincides with the monotone span program size of their access structure. Prior to this work there was no evidence that nonlinear schemes can be significantly more efficient than linear schemes, and in particular there were no candidates for schemes efficiently realizing access structures which do not lie in NC.
Fast parallel algorithms for matrix reduction to normal forms
 IN ENGINEERING, COMMUNICATION AND CONTROL
, 1997
"... We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in M (K), where K is an arbitrary commutative field, we establish that computing a similarity transformation P such that F"P��BP is in Frobenius normal form can be done ..."
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Cited by 4 (2 self)
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We investigate fast parallel algorithms to compute normal forms of matrices and the corresponding transformations. Given a matrix B in M (K), where K is an arbitrary commutative field, we establish that computing a similarity transformation P such that F"P��BP is in Frobenius normal form can be done in NC�. Using a reduction to this first problem, a similar fact is then proved for the Smith normal form S(x) of a polynomial matrix A(x) inM (K[x]); to compute unimodular matrices º(x) and »(x) such that S(x)"º(x)A(x)»(x) can be done in NC�. We get that over concrete fields such as the rationals, these problems are in NC². Using our previous results we have thus established that the problems of computing transformations over a field extension for the Jordan normal form, and transformations over the input field for the Frobenius and the Smith normal form are all in NC�. As a corollary we establish a polynomialtime sequential algorithm to compute transformations for the Smith form over K[x].