eng
Schloss Dagstuhl β Leibniz-Zentrum fΓΌr Informatik
Leibniz International Proceedings in Informatics
1868-8969
2022-07-11
25:1
25:28
10.4230/LIPIcs.CCC.2022.25
article
Finding Errorless Pessiland in Error-Prone Heuristica
Hirahara, Shuichi
1
Nanashima, Mikito
2
National Institute of Informatics, Tokyo, Japan
Tokyo Institute of Technology, Japan
Average-case complexity has two standard formulations, i.e., errorless complexity and error-prone complexity. In average-case complexity, a critical topic of research is to show the equivalence between these formulations, especially on the average-case complexity of NP.
In this study, we present a relativization barrier for such an equivalence. Specifically, we construct an oracle relative to which NP is easy on average in the error-prone setting (i.e., DistNP β HeurP) but hard on average in the errorless setting even by 2^o(n/log n)-size circuits (i.e., DistNP β AvgSIZE[2^o(n/log n)]), which provides an answer to the open question posed by Impagliazzo (CCC 2011). Additionally, we show the following in the same relativized world:
- Lower bound of meta-complexity: GapMINKT^πͺ β prSIZE^πͺ[2^o(n/log n)] and GapMCSP^πͺ β prSIZE^πͺ[2^(n^Ξ΅)] for some Ξ΅ > 0.
- Worst-case hardness of learning on uniform distributions: P/poly is not weakly PAC learnable with membership queries on the uniform distribution by nonuniform 2βΏ/n^Ο(1)-time algorithms.
- Average-case hardness of distribution-free learning: P/poly is not weakly PAC learnable on average by nonuniform 2^o(n/log n)-time algorithms.
- Weak cryptographic primitives: There exist a hitting set generator, an auxiliary-input one-way function, an auxiliary-input pseudorandom generator, and an auxiliary-input pseudorandom function against SIZE^πͺ[2^o(n/log n)].
This provides considerable insights into Pessiland (i.e., the world in which no one-way function exists, and NP is hard on average), such as the relativized separation of the error-prone average-case hardness of NP and auxiliary-input cryptography. At the core of our oracle construction is a new notion of random restriction with masks.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol234-ccc2022/LIPIcs.CCC.2022.25/LIPIcs.CCC.2022.25.pdf
average-case complexity
oracle separation
relativization barrier
meta-complexity
learning
auxiliary-input cryptography