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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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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.
Variations by complexity theorists on three themes of Euler, . . .
 COMPUTATIONAL COMPLEXITY
, 2005
"... This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and pa ..."
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Cited by 11 (3 self)
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This paper surveys some connections between geometry and complexity. A main role is played by some quantities —degree, Euler characteristic, Betti numbers — associated to algebraic or semialgebraic sets. This role is twofold. On the one hand, lower bounds on the deterministic time (sequential and parallel) necessary to decide a set S are established as functions of these quantities associated to S. The optimality of some algorithms is obtained as a consequence. On the other hand, the computation of these quantities gives rise to problems which turn out to be hard (or complete) in different complexity classes. These two kind of results thus turn the quantities above into measures of complexity in two quite different ways.
POLYNOMIAL HIERARCHY, BETTI NUMBERS AND A REAL ANALOGUE OF TODA’S THEOREM
"... Abstract. Toda [35] proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #P, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity cla ..."
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Cited by 8 (5 self)
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Abstract. Toda [35] proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #P, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of BlumShubSmale real Turing machines [9]) has been missing so far. In this paper we formulate and prove a real analogue of Toda’s theorem. Unlike Toda’s proof in the discrete case, which relied on sophisticated combinatorial arguments, our proof is topological in nature. As a consequence of our techniques we are also able to relate the computational hardness of two extremely wellstudied problems in algorithmic semialgebraic geometry – namely the problem of deciding sentences in the first order theory of the reals with a constant number of quantifier alternations, and that of computing Betti numbers of semialgebraic sets. We obtain a polynomial time reduction of the compact version of the first problem to the second. This latter result might be of independent interest to researchers in algorithmic semialgebraic geometry.
On the complexity of deciding connectedness and computing Betti numbers of a complex algebraic variety
 J. Complexity
"... We extend the lower bounds on the complexity of computing Betti numbers proved in [6] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACEhard. Then ..."
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Cited by 6 (2 self)
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We extend the lower bounds on the complexity of computing Betti numbers proved in [6] to complex algebraic varieties. More precisely, we first prove that the problem of deciding connectedness of a complex affine or projective variety given as the zero set of integer polynomials is PSPACEhard. Then we prove PSPACEhardness for the more general problem of deciding whether the Betti number of fixed order of a complex affine or projective variety is at most some given integer. Key words: connected components, Betti numbers, PSPACE, lower bounds 1
COUNTING COMPLEXITY CLASSES FOR NUMERIC COMPUTATIONS I: SEMILINEAR SETS
"... We define a counting class #Padd in the BlumShubSmalesetting of additive computations over the reals. Structural properties of this class are studied, including a characterization in terms of the classical counting class #P introduced by Valiant. We also establish transfer theorems for both dire ..."
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Cited by 5 (4 self)
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We define a counting class #Padd in the BlumShubSmalesetting of additive computations over the reals. Structural properties of this class are studied, including a characterization in terms of the classical counting class #P introduced by Valiant. We also establish transfer theorems for both directions between the real additive and the discrete setting. Then we characterize in terms of completeness results the complexity of computing basic topological invariants of semilinear sets given by additive circuits. It turns out that the computation of the Euler characteristic is FP #Padd addcomplete, while for fixed k, the computation of the kth Betti number is FPARaddcomplete. Thus the latter is more difficult under standard complexity theoretic assumptions. We use all the above to prove some analogous completeness results in the classical setting.
The complexity of computing the Hilbert polynomial of smooth equidimensional complex projective varieties
, 2005
"... We continue the study of counting complexity begun in [7, 8, 9] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be r ..."
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Cited by 4 (2 self)
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We continue the study of counting complexity begun in [7, 8, 9] by proving upper and lower bounds on the complexity of computing the Hilbert polynomial of a homogeneous ideal. We show that the problem of computing the Hilbert polynomial of a smooth equidimensional complex projective variety can be reduced in polynomial time to the problem of counting the number of complex common zeros of a finite set of multivariate polynomials. Moreover, we prove that the more general problem of computing the Hilbert polynomial of a homogeneous ideal is polynomial space hard. This implies polynomial space lower bounds for both the problems of computing the rank and the Euler characteristic of cohomology groups of coherent sheaves on projective space, improving the #Plower bound in Bach [1].
A Numerical Algorithm for Zero Counting. I: Complexity and Accuracy
, 2008
"... We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(log(nDκ(f))) iterations (grid refinements) where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials ’ degree, ..."
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Cited by 4 (2 self)
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We describe an algorithm to count the number of distinct real zeros of a polynomial (square) system f. The algorithm performs O(log(nDκ(f))) iterations (grid refinements) where n is the number of polynomials (as well as the dimension of the ambient space), D is a bound on the polynomials ’ degree, and κ(f) is a condition number for the system. Each iteration uses an exponential number of operations. The algorithm uses finiteprecision arithmetic and a major feature in our results is a bound for the precision required to ensure the returned output is correct which is polynomial in n and D and logarithmic in κ(f). The algorithm parallelizes well in the sense that each iteration can be computed in parallel time polynomial in n, logD and log(κ(f)).
On the Complexity of Counting Components of Algebraic Varieties
, 2008
"... We give a uniform method for the two problems of counting the connected and irreducible components of complex algebraic varieties. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work in parallel polynomial time, i.e., they can be implemented by algebraic circ ..."
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Cited by 3 (2 self)
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We give a uniform method for the two problems of counting the connected and irreducible components of complex algebraic varieties. Our algorithms are purely algebraic, i.e., they use only the field structure of C. They work in parallel polynomial time, i.e., they can be implemented by algebraic circuits of polynomial depth. The design of our algorithms relies on the concept of algebraic differential forms. A further important building block is an algorithm of Szántó computing a variant of characteristic sets. Furthermore, we use these methods to obtain a parallel polynomial time algorithm for computing the Hilbert polynomial of a projective variety which is arithmetically CohenMacaulay.
A COMPLEX ANALOGUE OF TODA’S THEOREM
"... Abstract. Toda [24] proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #P, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity cla ..."
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Cited by 3 (1 self)
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Abstract. Toda [24] proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #P, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result (with a compactness hypothesis) in the complexity theory over the reals (in the sense of BlumShubSmale real machines [4]) was proved in [2]. Unlike Toda’s proof in the discrete case, which relied on sophisticated combinatorial arguments, the proof in [2] is topological in nature in which the properties of the topological join is used in a fundamental way. However, the constructions used in [2] were semialgebraic – they used real inequalities in an essential way and as such do not extend to the complex case. In this paper, we extend the techniques developed in [2] to the complex projective case. A key role is played by the complex join of quasiprojective complex varieties. As a consequence we obtain a complex analogue of Toda’s theorem. The results contained in this paper, taken together with those contained in [2], illustrate the central role of the Poincaré polynomial in algorithmic algebraic geometry, as well as, in computational complexity theory over the complex and real numbers – namely, the ability to compute it efficiently enables one to decide in polynomial time all languages in the (compact) polynomial hierarchy over the appropriate field.
A COMPLEXITY THEORY OF CONSTRUCTIBLE FUNCTIONS AND SHEAVES
"... Abstract. In this paper we introduce analogues of the discrete complexity classes VP and VNP of sequences of functions in the BlumShubSmale model. The functions in the new definitions are constructible functions on Rn. We define a class of sequences of constructible functions that play a role anal ..."
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Abstract. In this paper we introduce analogues of the discrete complexity classes VP and VNP of sequences of functions in the BlumShubSmale model. The functions in the new definitions are constructible functions on Rn. We define a class of sequences of constructible functions that play a role analogous to that of VP in the discrete theory. The class analogous of VNP is defined using Euler integration. We discuss several examples and pose a conjecture analogous to the VP vs VNP conjecture in the discrete case. In the second part of the paper we extend the notions of complexity classes to sequences of constructible sheaves over Rn (or its one point compactification). We introduce a class of sequences simple constructible sheaves, that could be seen as the sheaf theoretic analog of the BlumShubSmale class PR. We also define a hierarchy of complexity classes of sheaves mirroring the polynomial hierarchy, PHR in the more classical theory. We prove singly exponential upper bounds on the topological complexity of the sheaves in this hierarchy. We obtain as a result a singly exponential complexity upper bound on the sheaftheoretic version of quantifier elimination. We pose the natural sheaf theoretic analogues of the classical P vs NP question and also discuss a connection with Toda’s theorem from discrete complexity theory. 1.