For Complex Intervals, Exact Range Computation Is NP-Hard Even for Single Use Expressions (Even for the Product)

One of the main problems of interval computations is to compute the range y of the given function f(x1, . . . , xn) under interval uncertainty. Interval computations started with the invention of straightforward interval computations, when we simply replace each elementary arithmetic operation in the code for f with the corresponding operation from interval arithmetic. In general, this technique only leads to an enclosure Y ⊇ y for the desired range, but in the important case of single use expressions (SUE), in which each variable occurs only once, we get the exact range. Thus, for SUE expressions, there exists a feasible (polynomial-time) algorithm for computing the exact range. We show that in the complex-valued case, computing the exact range is NP-hard even for SUE expressions. Moreover, it is NP-hard even for such simple expressions as the product f(z1, . . . , zn) = z1 · . . . · zn.