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On the Complexity of Numerical Analysis
Citation key ABKM-On-The-Complexity-Of-Numerical-Analysis
Author Eric Allender and Peter Bürgisser and Johan Kjeldgaard-Pedersen and Peter Bro Miltersen
Pages 1987-2006
Year 2009
Journal SIAM J. Comput.
Volume 38
Number 5
Abstract We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis. We show that both 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 $N>0$. We show that PosSLP lies in the counting hierarchy, and we show that if $A$ is any language in the Boolean part of $P_R$ accepted by a machine whose machine constants are algebraic real numbers, then $A \in $P^textPosSLP$. Combining our results with work of Tiwari, we show 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.
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