Improving on Gutfreund, Shaltiel,and Ta-Shma’s Paper “If NP Languages Are Hard on the Worst-Case, Then It Is Easy to Find Their Hard Instances”

Abstract

Assume that $\text{NP}\not\subset\text{BPP}$. Gutfreund, Shaltiel, and Ta-Shma in [Computational Complexity 16(4):412-441 (2007)] have proved that for every randomized polynomial time decision algorithm D for SAT there is a polynomial time samplable distribution such that D errs with probability at least 1/6 − ε on a random formula chosen with respect to that distribution. A challenging problem is to increase the error probability to the maximal possible 1/2 − ε (the random guessing has success probability 1/2). In this paper, we make a small step towards this goal: we show how to increase the error probability to 1/3 − ε.