Revision #1 Authors: Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, Peter Bro Miltersen

Accepted on: 6th December 2005 00:00

Downloads: 2113

Keywords:

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 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.

TR05-037 Authors: Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, Peter Bro Miltersen

Publication: 9th April 2005 09:08

Downloads: 2119

Keywords:

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 Polynomial-time

over the Reals accepted by a machine whose machine constants are

algebraic real numbers, then A is in P^PosSLP. 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.

Comment #1 Authors: Eric Allender, Peter Bürgisser, Johan Kjeldgaard-Pedersen, Peter Bro Miltersen

Accepted on: 26th April 2005 16:52

Downloads: 1477

Keywords:

We thank Klaus Meer for calling our attention to the fact that Theorem 1.2

follows very easily from Lemma 4.4 and Theorem 4.5 of the paper

``Saturation and Stability in the Theory of Computation over the Reals'',

by Olivier Chapuis and Pascal Koiran (Annals of Pure and Applied Logic

99 (1999) pp. 1-49). Their work shows more generally how algebraic

constants can be eliminated in computation over the reals.