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On the Complexity of Numerical Analysis

Authors: Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, and Peter Bro Miltersen

Published in: Dagstuhl Seminar Proceedings, Volume 6111, Complexity of Boolean Functions (2006)

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 OrdSlp lies in the counting hierarchy, and 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.

Cite as

Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, and Peter Bro Miltersen. On the Complexity of Numerical Analysis. In Complexity of Boolean Functions. Dagstuhl Seminar Proceedings, Volume 6111, pp. 1-9, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2006)

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  author =	{Allender, Eric and B\"{u}rgisser, Peter and Kjeldgaard-Pedersen, Johan and Miltersen, Peter Bro},
  title =	{{On the Complexity of Numerical Analysis}},
  booktitle =	{Complexity of Boolean Functions},
  pages =	{1--9},
  series =	{Dagstuhl Seminar Proceedings (DagSemProc)},
  ISSN =	{1862-4405},
  year =	{2006},
  volume =	{6111},
  editor =	{Matthias Krause and Pavel Pudl\'{a}k and R\"{u}diger Reischuk and Dieter van Melkebeek},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.06111.12},
  URN =		{urn:nbn:de:0030-drops-6130},
  doi =		{10.4230/DagSemProc.06111.12},
  annote =	{Keywords: Blum-Shub-Smale Model, Euclidean Traveling Salesman Problem, Counting Hierarchy}
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