The PCP-like Theorem for 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 proving sub-quadratic time inapproximability. Here we try to go further in this direction. Staring from SETH, we first find a problem denoted as Ext-$k$-SAT, which can not be computed in linear time, then devise an efficient MA-like protocol for this problem. To use this protocol to prove the sub-linear time inapproximability of other problems, we devise a new kind of reduction denoted as Ext-reduction, and it is different from existing reduction techniques. We also define two new hardness class, the problems in which can be computed in linear-time, but can not be efficiently approximated in sub-linear time. Some problems are shown to be in the newly defined hardness class.