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Hard-Core Distributions for Somewhat Hard Problems (1995)
| Venue: | In 36th Annual Symposium on Foundations of Computer Science |
| Citations: | 122 - 11 self |
BibTeX
@INPROCEEDINGS{Impagliazzo95hard-coredistributions,
author = {Russell Impagliazzo},
title = {Hard-Core Distributions for Somewhat Hard Problems},
booktitle = {In 36th Annual Symposium on Foundations of Computer Science},
year = {1995},
pages = {538--545},
publisher = {IEEE}
}
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Abstract
Consider a decision problem that cannot be 1 \Gamma ffi approximated by circuits of a given size in the sense that any such circuit fails to give the correct answer on at least a ffi fraction of instances. We show that for any such problem there is a specific "hard-core" set of inputs which is at least a ffi fraction of all inputs and on which no circuit of a slightly smaller size can get even a small advantage over a random guess. More generally, our argument holds for any non-uniform model of computation closed under majorities. We apply this result to get a new proof of the Yao XOR lemma [Y], and to get a related XOR lemma for inputs that are only k-wise independent. 1 Introduction If you have a difficult computational problem, is it always the case that several independent instances of the problem are proportionately harder than a single instance? In particular, if any algorithm taking less than R resources has failure probability at least ffi for a particular problem on a certai...
Keyphrases
somewhat hard problem hard-core distribution ffi fraction small advantage related xor lemma random guess specific hard-core set decision problem circuit fails failure probability gamma ffi non-uniform model new proof particular problem yao xor lemma correct answer single instance difficult computational problem several independent instance
