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On the Complexity of Numerical Analysis (Conference Version)
Citation key ABKM-On-The-Complexity-Of-Numerical-Analysis-Conference-Version
Author Eric Allender and Peter Bürgisser and Johan Kjeldgaard-Pedersen and Peter Bro Miltersen
Title of Book In Proc. of 21st Ann. IEEE Conference on Computational Complexity
Pages 331-339
Year 2006
Address Prague
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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