Abstract. We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis:• The Blum-Shub-Smale 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 division-free straight-line program producing an integer N , decide whether N > 0.• In the Blum-Shub-Smale model, polynomial time computation over the reals (on discrete inputs) is polynomial-time 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 polynomial-time 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.
The purpose of this overview is to explain the enormous impact of Les Valiant's eponymous short conference contribution from 1979 on the development of algebraic complexity. Contents 1. Introduction 2. Foundations 2.1. Arithmetic circuits 2.2. Valiant's complexity classes 2.3. Reduction via substitution 2.4. Circuits for the determinant 2.5. Completeness of the determinant 2.6. The class VNP 2.7. Completeness of the permanent 2.8. Generating functions of graph properties 2.9. Determinantal complexity 3. Properties of complexity classes 3.1. Robustness 3.2. Complexity of factors 3.3. Families of intermediate complexity 4. Second generation algebraic complexity classes 4.1. Constant-free classes 4.2. Non-bounded degree: the classes VP 0 nb and VNP 0 nb 4.3. The class VPSPACE 0 4.4. Connection to Blum-Shub-Smale model 4.5. Counting classes for algebraic geometry 4.6. Tau Conjectures 4.7. Closures of complexity classes 5. Restricted models of computation 5.1. Monotone circuits 5.2. Multilinear circuits 5.3. Set-multilinear circuits 5.4. Noncommutative model
Abstract. The 17th of the problems proposed by Steve Smale for the 21st century asks for the existence of a deterministic algorithm computing an approximate solution of a system of n complex polynomials in n unknowns in time polynomial, on the average, in the size N of the input system. A partial solution to this problem was given by Carlos Beltrán and Luis Miguel Pardo who exhibited a randomized algorithm doing so. In this paper we further extend this result in several directions. Firstly, we exhibit a linear homotopy algorithm that efficiently implements a non-constructive idea of Mike Shub. This algorithm is then used in a randomized algorithm, call it LV,à la Beltrán-Pardo. Secondly, we perform a smoothed analysis (in the sense of Spielman and Teng) of algorithm LV and prove that its smoothed complexity is polynomial in the input size and σ −1 , where σ controls the size of of the random perturbation of the input systems. Thirdly, we perform a condition-based analysis of LV. That is, we give a bound, for each system f , of the expected running time of LV with input f . In addition to its dependence on N this bound also depends on the condition of f . Fourthly, and to conclude, we return to Smale's 17th problem as originally formulated for deterministic algorithms. We exhibit such an algorithm and show that its average complexity is N O(log log N) . This is nearly a solution to Smale's 17th problem.
We define counting classes #P R and #P C in the Blum-Shub-Smale setting of computations over the real or complex numbers, respectively. The problems of counting the number of solutions of systems of polynomial inequalities over R, or of systems of polynomial equalities over C, respectively, turn out to be natural complete problems in these classes. We investigate to what extent the new counting classes capture the complexity of computing basic topological invariants of semialgebraic sets (over R) and algebraic sets (over C). We prove that the problem of computing the Euler-Yao characteristic of semialgebraic sets is FP #P R R -complete, and that the problem of computing the geometric degree of complex algebraic sets is FP #P C C -complete. We also define new counting complexity classes in the classical Turing model via taking Boolean parts of the classes above, and show that the problems to compute the Euler characteristic and the geometric degree of (semi)algebraic sets given by integer polynomials are complete in these classes. We complement the results in the Turing model by proving, for all k ∈ N, the FPSPACE-hardness of the problem of computing the kth Betti number of the set of real zeros of a given integer polynomial. This holds with respect to the singular homology as well as for the Borel-Moore homology.
We discuss the geometry of orbit closures and the asymptotic behavior of Kronecker coefficients in the context of the Geometric Complexity Theory program to prove a variant of Valiant's algebraic analog of the P = N P conjecture. We also describe the precise separation of complexity classes that their program proposes to demonstrate.
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