On the computational complexity of best Chebyshev approximations

Abstract

The best Chebyshev approximation of degree n to a continuous function f on [0, 1] is the unique polynomial ϕ of degree less than or equal to n such that the maximum difference of f and ϕ on [0, 1] is minimized. On the basis of a formal model of computation, it is shown that the question of whether the best Chebyshev approximations of polynomial-time computable functions on [0, 1] are always polynomial-time computable depends on the relationship among well-known discrete complexity classes. In particular, P = NP implies that these best approximations are polynomial-time computable, and EXP ≠ NEXP implies that these best approximations are not polynomial-time computable. It is also pointed out that the fact that the popular Remes algorithm converges fast does not conflict with the above result, since the Remes algorithm requires, in each iteration, the finding of maximal points of continuous functions on an interval [ a, b], which is, in general, provably intractable.