Title: On the Complexity of Numerical Analysis

Speaker: **Eric Allender**, Rutgers University

Date: December 13, 2005 2:00-3:00pm

Location: DIMACS Center, CoRE Bldg, Room 431, Rutgers University, Busch Campus, Piscataway, NJ

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^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.

Joint work with Peter B"urgisser, Johan Kjeldgaard-Pedersen, and Peter Bro Miltersen.