On the Approximation Method and the P versus NP Problem

TL;DR

This paper examines the limitations of the approximation method in proving P ≠ NP, providing evidence that it cannot establish super-polynomial lower bounds for Boolean function complexities and discussing future research directions.

Contribution

It analyzes the approximation method's potential and limitations in proving super-polynomial lower bounds for Boolean functions in NP, offering insights and personal opinions on future research.

Findings

Approximation method alone cannot prove super-linear lower bounds.

Natural proofs are not a barrier to proving P ≠ NP.

Overview of methods for lower bounds in Boolean function complexity.

Abstract

First of all we give some reasons that "natural proofs" built not a barrier to prove P $\not=$ NP using Boolean complexity. Then we investigate the approximation method for its extension to prove super-polynomial lower bounds for the non-monotone complexity of suitable Boolean functions in NP or to understand why this is not possible. It is given some evidence that the approximation method alone cannot be used to prove a super-linear lower bound for any function $f \in {\cal B}_n$. Additionally, an overview on the methods for proving lower bounds of the non-monotone and the monotone complexity of Boolean functions is given. Finally, a personal opinion how to proceed the research on the P versus NP problem and also on proving a super-linear lower bound for the non-monotone complexity of a Boolean function in NP is given.