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19
On the complexity of numerical analysis
 IN PROC. 21ST ANN. IEEE CONF. ON COMPUTATIONAL COMPLEXITY (CCC ’06
, 2006
"... We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation ..."
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Cited by 46 (6 self)
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We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: • The BlumShubSmale model of computation over the reals. • A problem we call the “Generic Task of Numerical Computation, ” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model ” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a divisionfree straightline program producing an integer N, decide whether N> 0. • In the BlumShubSmale model, polynomial time computation over the reals (on discrete inputs) is polynomialtime equivalent to PosSLP, when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. • The Generic Task of Numerical Computation is also polynomialtime equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean Traveling Salesman Problem lies in the counting hierarchy – the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of ACIT: the Arithmetic Circuit Identity Testing problem. In particular, we show that if n! is not ultimately easy, then ACIT has subexponential complexity.
Arithmetic circuits and counting complexity classes
 In Complexity of Computations and Proofs,J.Krajíček, Ed. Quaderni di Matematica
"... Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in ..."
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Cited by 20 (3 self)
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Arithmetic circuits are the focus of renewed attention in the complexity theory community. It is easy to list a few of the reasons for the increased interest: • Innovative work by Kabanets and Impagliazzo [KI03] shows that, in
Nonuniform ACC circuit lower bounds
, 2010
"... The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynom ..."
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Cited by 19 (4 self)
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The class ACC consists of circuit families with constant depth over unbounded fanin AND, OR, NOT, and MODm gates, where m> 1 is an arbitrary constant. We prove: • NTIME[2 n] does not have nonuniform ACC circuits of polynomial size. The size lower bound can be slightly strengthened to quasipolynomials and other less natural functions. • ENP, the class of languages recognized in 2O(n) time with an NP oracle, doesn’t have nonuniform ACC circuits of 2no(1) size. The lower bound gives an exponential sizedepth tradeoff: for every d there is a δ> 0 such that ENP doesn’t have depthd ACC circuits of size 2nδ. Previously, it was not known whether EXP NP had depth3 polynomial size circuits made out of only MOD6 gates. The highlevel strategy is to design faster algorithms for the circuit satisfiability problem over ACC circuits, then prove that such algorithms entail the above lower bounds. The algorithm combines known properties of ACC with fast rectangular matrix multiplication and dynamic programming, while the second step requires a subtle strengthening of the author’s prior work [STOC’10]. Supported by the Josef Raviv Memorial Fellowship.
Bounded Depth Arithmetic Circuits: Counting and Closure
, 1999
"... Constantdepth arithmetic circuits have been defined and studied in [AAD97, ABL98]; these circuits yield the function classes #AC . These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC ..."
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Cited by 10 (3 self)
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Constantdepth arithmetic circuits have been defined and studied in [AAD97, ABL98]; these circuits yield the function classes #AC . These function classes in turn provide new characterizations of the computational power of threshold circuits, and provide a link between the circuit classes AC (where many lower bounds are known) and TC (where essentially no lower bounds are known). In this paper, we resolve several questions regarding the closure properties of #AC .
Unary Quantifiers, Transitive Closure, and Relations of Large Degree
"... This paper studies expressivity bounds for extensions of firstorder logic with counting and unary quantifiers in the presence of relations of large degree. There are several motivations for this work. First, it is known that firstorder logic with counting quantifiers captures uniform TC 0 over ..."
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Cited by 8 (4 self)
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This paper studies expressivity bounds for extensions of firstorder logic with counting and unary quantifiers in the presence of relations of large degree. There are several motivations for this work. First, it is known that firstorder logic with counting quantifiers captures uniform TC 0 over ordered structures. Thus, proving expressivity bounds for firstorder with counting can be seen as an attempt to show TC 0 $ DLOG using techniques of descriptive complexity. Second, the presence of auxiliary builtin relations (e.g., order, successor) is known to make a big impact on expressivity results in finitemodel theory and database theory (where logics with counting and unary quantifiers have recently been used to model query languages with aggregation). For those logics, our goal is to extend techniques from "pure" setting to that of auxiliary relations. Until now, all known results on the limitations of expressive power of the counting and unary quantifier extensions of firstorder...
Arithmetic Circuits and Polynomial Replacement Systems
 the Proceedings of the 2000 FSTTCS conference
, 1999
"... This paper addresses the problems of counting proof trees (as introduced by Venkateswaran and Tompa) and counting proof circuits, a related but seemingly more natural question. These problems lead to a common generalization of straightline programs which we call polynomial replacement systems. W ..."
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Cited by 5 (2 self)
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This paper addresses the problems of counting proof trees (as introduced by Venkateswaran and Tompa) and counting proof circuits, a related but seemingly more natural question. These problems lead to a common generalization of straightline programs which we call polynomial replacement systems. We contribute a classication of these systems and we investigate their complexity. Diverse problems falling in the scope of this study include, for example, counting proof circuits, and evaluating f[; +gcircuits over the natural numbers. A number of complexity results are obtained, e.g., the former problem is shown #Pcomplete, while the latter is shown to be equivalent to a particular type of replacement systems. 1 Introduction 1.1 Motivation ^ x1 x2 g2 g1 _ g3 _ _ g4 When + and replace _ and ^ in the adjacent gure, the gate g 1 on input x 1 = x 2 = 1 evaluates to 9. Equivalently, the treelike Boolean circuit T obtained from the circuit drawn has 9 proof trees [VT89],...
On the Complexity of Mining Association Rules
, 2001
"... Fabrizio Angiulli 1 , Giovambattista Ianni 2 , and Luigi Palopoli 3 1 ISICNR c/o Universita della Calabria, DEIS, Via P. Bucci 41C, Rende, Italy angiulli@isi.cs.cnr.it 2 Universita della Calabria, DEIS, Via P. Bucci 41C, Rende, Italy ianni@deis.unical.it 3 Universita di Reggio Calabria, DIMET, Loc. ..."
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Cited by 5 (0 self)
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Fabrizio Angiulli 1 , Giovambattista Ianni 2 , and Luigi Palopoli 3 1 ISICNR c/o Universita della Calabria, DEIS, Via P. Bucci 41C, Rende, Italy angiulli@isi.cs.cnr.it 2 Universita della Calabria, DEIS, Via P. Bucci 41C, Rende, Italy ianni@deis.unical.it 3 Universita di Reggio Calabria, DIMET, Loc. Feo di Vito, Reggio Calabria, Italy palopoli@ing.unirc.it Abstract. In this paper we describe our ongoing research towards establishing the complexity of mining association rules from relational databases. We consider both quantitative, categorical and boolean association rules and various forms of quality indexes, including confidence, support, gain, laplace. The presented results show that all these problems are, generally, computationally hard to solve, even if we are able to single out some interesting tractable special cases.
Cracks in the Defenses: Scouting Out Approaches on Circuit Lower Bounds
"... Razborov and Rudich identified an imposing barrier that stands in the way of progress toward the goal of proving superpolynomial lower bounds on circuit size. Their work on “natural proofs” applies to a large class of arguments that have been used in complexity theory, and shows that no such argum ..."
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Cited by 4 (1 self)
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Razborov and Rudich identified an imposing barrier that stands in the way of progress toward the goal of proving superpolynomial lower bounds on circuit size. Their work on “natural proofs” applies to a large class of arguments that have been used in complexity theory, and shows that no such argument can prove that a problem requires circuits of superpolynomial size, even for some very restricted classes of circuits (under reasonable cryptographic assumptions). This barrier is so daunting, that some researchers have decided to focus their attentions elsewhere. Yet the goal of proving circuit lower bounds is of such importance, that some in the community have proposed concrete strategies for surmounting the obstacle. This lecture will discuss some of these strategies, and will dwell at length on a recent approach proposed by Michal Koucky and the author.
On the power of algebraic branching programs of width two
"... Abstract. We show that there are families of polynomials having small depthtwo arithmetic circuits that cannot be expressed by algebraic branching programs of width two. This clarifies the complexity of the problem of computing the product of a sequence of twobytwo matrices, which arises in sever ..."
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Cited by 3 (1 self)
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Abstract. We show that there are families of polynomials having small depthtwo arithmetic circuits that cannot be expressed by algebraic branching programs of width two. This clarifies the complexity of the problem of computing the product of a sequence of twobytwo matrices, which arises in several settings. 1
On the Complexity of Inducing Categorical and Quantitative Association Rules
, 2001
"... Inducing association rules is one of the central tasks in data mining applications. Quantitative association rules induced from databases describe rich and hidden relationships holding within data that can prove useful for various application purposes (e.g., market basket analysis, customer profilin ..."
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Cited by 3 (0 self)
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Inducing association rules is one of the central tasks in data mining applications. Quantitative association rules induced from databases describe rich and hidden relationships holding within data that can prove useful for various application purposes (e.g., market basket analysis, customer profiling, and others). Even though such association rules are quite widely used in practice, a thorough analysis of the computational complexity of inducing them is missing. This paper intends to provide a contribution in this setting. To this end, we first formally define quantitative association rule mining problems, which entail boolean association rules as a special case, and then analyze their computational complexities, by considering both the standard cases, and a some special interesting case, that is, association rule induction over databases with null values, fixedsize attribute set databases, sparse databases, fixed threshold problems. 1