On optimal polynomial time approximations: P-levelability vs. δ-levelability

Abstract

Safe and unsafe polynomial time approximations were introduced by Meyer and Paterson [4] and Yesha [8], respectively. The question of which sets have optimal safe approximations was investigated by several authors (see e.g. [3,6,7]). Orponen et al. [5] showed that a problem has an optimal polynomial time approximation if and only if neither it nor its complement is p-levelable. Recently Duris and Rolim [2] considered the unsafe case and compared the existence of optimal polynomial time approximations for both cases. They left open the question, however, whether there are intractable sets with optimal unsafe approximations and whether there are sets with optimal unsafe approximations but without optimal safe approximations. Here we answer these questions affirmatively. Moreover, we consider a variant of Duris and Rolim's δ-levelability concept related to the nonexistence of optimal unsafe approximations.

This work was supported in part by the HCM program of the European Community under grant CHRX-CT93-0415 (COLORET Network).