AverageCase Complexity
Abstract
We survey the averagecase complexity of problems in NP. We discuss various notions of goodonaverage 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 easyonaverage with respect to the uniform distribution, then all problems in NP are easyonaverage 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 averagecase complexity is of particular interest and that do not yet fit into this theory. A major open question whether the existence of hardonaverage problems in NP can be based on the P$\neq$NP assumption or on related worstcase assumptions. We review negative results showing that certain proof techniques cannot prove such a result. While the relation between worstcase and averagecase complexity for general NP problems remains open, there has been progress in understanding the relation between different "degrees" of averagecase complexity. We discuss some of these "hardness amplification" results.
 Publication:

arXiv eprints
 Pub Date:
 June 2006
 arXiv:
 arXiv:cs/0606037
 Bibcode:
 2006cs........6037B
 Keywords:

 Computer Science  Computational Complexity