PCP Theorems, SETH and More: Towards Proving Sub-linear Time Inapproximability

TL;DR

This paper introduces a sub-linear time PCP theorem that extends previous frameworks to prove inapproximability results for polynomial time algorithms, highlighting its significance in the development of sub-linear time approximation algorithms.

Contribution

The paper generalizes the distributed PCP framework to sub-linear time, proving a new sub-linear PCP theorem and demonstrating its application to existing and novel inapproximability results.

Findings

Established a sub-linear PCP theorem using an MA-protocol for Set Containment.

Proved the theorem's limitations in the linear case.

Applied the theorem to derive new inapproximability results.

Abstract

In this paper we propose the PCP-like theorem for sub-linear time inapproximability. Abboud et al. have devised the distributed PCP framework for sub-quadratic time inapproximability. We show that the distributed PCP theorem can be generalized for proving arbitrary polynomial time inapproximability, but fails in the linear case. We prove the sub-linear PCP theorem by adapting from an MA-protocol for the Set Containment problem, and show how to use the theorem to prove both existing and new inapproximability results, exhibiting the power of the sub-linear PCP theorem. Considering the emerging research works on sub-linear time algorithms, the sub-linear PCP theorem is important in guiding the research in sub-linear time approximation algorithms.