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Average-Case Complexity
Author(s): Andrej Bogdanov;Luca Trevisan
Source:
Journal:Foundations and Trends® in Theoretical Computer Science
ISSN Print:1551-305X, ISSN Online:1551-3068
Publisher:Now Publishers
Volume 2 Number 1,
Document Type: Article
Pages: 106(1-106)
DOI: 10.1561/0400000004
Abstract: We survey the average-case complexity of problems in NP. We discuss various notions of good-on-average algorithms, and present completeness results due to Impagliazzo and Levin. Such completeness results establish the fact that if a certain specific (but somewhat artificial) NP problem is easy-on-average with respect to the uniform distribution, then all problems in NP are easy-on-average with respect to all samplable distributions. Applying the theory to natural distributional problems remain an outstanding open question. We review some natural distributional problems whose average-case complexity is of particular interest and that do not yet fit into this theory. A major open question is whether the existence of hard-on-average problems in NP can be based on the P ≠ NP assumption or on related worst-case assumptions. We review negative results showing that certain proof techniques cannot prove such a result. While the relation between worst-case and average-case complexity for general NP problems remains open, there has been progress in understanding the relation between different "degrees" of average-case complexity. We discuss some of these "hardness amplification" results.
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