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.
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.
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.
