NDLI: On WorstCase to AverageCase Reductions for NP Problems

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On WorstCase to AverageCase Reductions for NP Problems

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Bogdanov, Andrej Trevisan, Luca
Copyright Year	2006
Abstract	We show that if an NPcomplete problem has a nonadaptive selfcorrector with respect to any samplable distribution, then coNP is contained in NP/poly and the polynomial hierarchy collapses to the third level. Feigenbaum and Fortnow SIAM J. Comput., 22 1993, pp. 9941005 show the same conclusion under the stronger assumption that an NPcomplete problem has a nonadaptive random selfreduction. A selfcorrector for a language L with respect to a distribution $\cal D$ is a worstcase to averagecase reduction that transforms any given algorithm that correctly decides L on most inputs with respect to $\cal D$ into an algorithm of comparable efficiency that decides L correctly on every input. A random selfreduction is a special case of a selfcorrector, where the reduction, given an input x, is restricted to only making oracle queries that are distributed according to $\cal D$. The result of Feigenbaum and Fortnow depends essentially on the property that the distribution of each query in a random selfreduction is independent of the input of the reduction. Our result implies that the averagecase hardness of a problem in NP or the security of a oneway function cannot be based on the worstcase complexity of an NPcomplete problem via nonadaptive reductions unless the polynomial hierarchy collapses.
Starting Page	1119
Ending Page	1159
Page Count	41
File Format	PDF
ISSN	00975397
DOI	10.1137/S0097539705446974
e-ISSN	10957111
Journal	SIAM Journal on Computing (SMJCAT)
Issue Number	4 (Special Issue on Randomness and Complexity)
Volume Number	36
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-12-15
Access Restriction	Subscribed
Subject Keyword	averagecase complexity worstcase to averagecase reduction oneway function
Content Type	Text
Resource Type	Article
Subject	Mathematics Computer Science

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