Hardness amplification for errorless heuristics

An errorless heuristic is an algorithm that on all inputs returns either the correct answer or the special symbol ⊥, which means "I don't know." A central question in average-case complexity is whether every distributional decision problem in NP has an errorless heuristic scheme: This is an algorithm that, for every δ > 0, runs in time polynomial in the instance size and 1/δ and answers ⊥. only on a δ fraction of instances. We study the question from the standpoint of hardness amplification and show that If every problem in (NP, u) has errorless heuristic circuits that output the correct answer on n^-2/9+o(1)-fraction of inputs, then (NP, u) has non-uniform errorless heuristic schemes. If every problem in (NP, u) has randomized errorless heuristic algorithms that output the correct answer on (log n)^-1/10+o(1)-fraction of inputs, then (NP, u) has randomized errorless heuristic schemes. In both cases, the low-end amplification is achieved by analyzing a new sensitivity property of monotone boolean functions in NP. In the non-uniform setting we use a "holographic function" introduced by Benjamini, Schramm, and Wilson (STOC 2005). For the uniform setting we introduce a new function that can be viewed as an efficient version of Talagrand's "random DNF".