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

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.