E-complete sets do not have optimal polynomial time approximations

Abstract

The p-levelability of intractable sets that are, for example, honestly paddable, or monotone self-reducible in certain restricted way, or complete for deterministic time classes is known, more particularly, all E-complete sets are p-levelable[5]. However, the extension of these results to a more general “unsafe” approximation notion has remained as a challenge for a few years. In this paper we address these questions; our main result can be stated as follows. Let k be any natural number. Then each E-complete set is levelable as well as Δ-levelable with density n ^k. This result extends the corresponding result mentioned above to “unsafe” approximation model and moreover it characterizes “density” of levelability for both models, “safe” and “unsafe” as well. In this paper, we also compare levelability with Δ-levelability and we show in this context that there are sets A, B and C in E-P such that A is levelable as well as Δ-levelable with exponential density, B is not levelable but it is Δ-levelable with exponential density, and C is not levelable and if C is Δ-levelable with a density f(n) then f(n) ≤log^* n + 1 for all n. Our proof technique is not based on paddability — it is based on diagonalization against all algorithms and polynomial time bounds.

This work was supported in part by the Swiss National Science Foundation grant number 21-32147-91