Optimization is one of the most common problems in both discrete and continuous computation. In discrete complexity theory, close connections between optimization and nondeterminism have been observed, as many combinatorial optimization problems have been shown to be NP-complete (cf. Garey and Johnson [1979]). On the other hand, optimization of a continuous function can often be solved in polynomial time. As an example, the linear programming problem is solvable in polynomial time [Khachiyan, 1979; Karmarkar, 1984], while the linear integer programming problem remains NP-hard. The question arises as to whether the continuous optimization problems are always easier than the discrete versions. In this chapter, we investigate this question on polynomial-time computable functions; more precisely, we consider the computational complexity of the maximum value of a polynomial-time computable function f on [0,1]. It is to be shown that these maximum values axe exactly the real numbers which have a (general) left cut in NP (called left NP real numbers). For two-dimensional, polynomial-time computable functions f on [0, l]2, the maximum functions g(x) = max{f(x, y)| 0 ≤ y ≤ 1} coincide with NP real functions (real functions whose undergraphs are in NP).


Polynomial Time Real Function Maximum Point Computable Function Binary Expansion 


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Copyright information

© Birkhäuser Boston 1991

Authors and Affiliations

  • Ker-I Ko
    • 1
  1. 1.Department of Computer ScienceState University of New York at Stony BrookStony BrookUSA

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