On the Computational Complexity of the Range Computation Problem

Abstract

The approximate range computation problem is one of the basic problem of interval computations. It is well known that this problem and even some special cases of it are at least as hard as any $\mathbf {NP}$-problem. First, we show that the general approximate range computation problem is not harder than $\mathbf {NP}$-problems. Then we show that the computional complexity of some further variants of this problem is closely related to some well-known open questions from structural complexity theory that seem to be slightly weaker than the famous open question whether the complexity class $\mathbf {NP}$ is equal to the complexity class $\mathbf {P}$, namely to the question whether $\mathbf {NE}$ is equal to $\mathbf {E}$, to the question whether $\mathbf {NEXP}$ is equal to $\mathbf {EXP}$ and, finally, to the question whether every $\mathbf {NP}$-real number is polynomial time computable.