Average-Case Complexity
<|reference_start|>Average-Case Complexity: 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 whether the existence of hard-on-average problems in NP can be based on the P$\neq$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.<|reference_end|>
@article{bogdanov2006average-case,
    title={Average-Case Complexity},
    author={Andrej Bogdanov and Luca Trevisan},
    journal={arXiv preprint arXiv:cs/0606037},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0606037},
    primaryClass={cs.CC}
}