Title
Finding the maximum of a polynomial time computable bounded smooth
function on an interval is NP-complete
Author
Warren D. Smith
Abstract
Ker-I Ko in his 1991 book ``Complexity theory of real functions''
(Birkhauser) defined the notions of polynomial-time computable
(P.T.C.) real numbers and real functions. On p.106 he posed as an open
question ``whether differentiability helps in computing maximum
values.'' We argue the answer is ``no.''
But in order to make this argument, Ko's theory needs to be extended
to encompass notions of the {\em codelength} of a P.T.C. real number,
and the {\em code transformation time} for various real arithmetic
operations. The question in Ko's unextended theory remains unsolved.
Our result is:
If $f(x)$ is $C^\infty$ and bounded and P.T.C. on $[0,1]$,
then the maximum $M$ of $f$ on $[0,1]$
either
(1) is not P.T.C.,
(2) or if it is, then the code of $M$ is not produceable
by an polytime algorithm from the code for $f$, or
(3) P=NP.
Keywords
maximization, minimization, polynomial time computable real numbers,
NP-completeness, code length.