Average-case complexity: Difference between revisions

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In 1990, Impagliazzo and Levin showed that if there is an efficient average-case algorithm for a distNP-complete problem under the uniform distribution, then there is an average-case algorithm for every problem in NP under any polynomial-time samplable distribution.<ref name="imp90">R. Impagliazzo and L. Levin, "No Better Ways to Generate Hard NP Instances than Picking Uniformly at Random," in Proceedings of the 31st IEEE Sympo- sium on Foundations of Computer Science, pp. 812–821, 1990.</ref> Applying this theory to natural distributional problems remains an outstanding open question.<ref name="bog06"/>

In 1992, Ben-David et al., showed that if all languages in distNP have good-on-average decision algorithms, they also have good-on-average search algorithms. Further, they show that this conclusion holds under a weaker assumption: if every language in NP is easy on average for decision algorithms with respect to the uniform distribution, then it is also easy on average for search algorithms with respect to the uniform distribution.<ref name="bd92">S. Ben-David, B. Chor, O. Goldreich, and M. Luby, "On the theory of average case complexity," Journal of Computer and System Sciences, vol. 44, no. 2, pp. 193–219, 1992.</ref> Thus, cryptographic one-way functions can exist only if there are distNP problems over the uniform distribution that are hard on average for decision algorithms.

In 1993, Feigenbaum and Fortnow showed that it is not possible to prove, under non-adaptive random reductions, that the existence of a good-on-average algorithm for a distNP-complete problem under the uniform distribution implies the existence of worst-case efficient algorithms for all problems in NP.<ref name="ff93">J. Feigenbaum and L. Fortnow, "Random-self-reducibility of complete sets," SIAM Journal on Computing, vol. 22, pp. 994–1005, 1993.</ref> In 2003, Bogdanov and Trevisan generalized this result to arbitrary non-adaptive reductions.<ref name="bog03">A. Bogdanov and L. Trevisan, "On worst-case to average-case reductions for NP problems," in Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, pp. 308–317, 2003.</ref> These results show that it is unlikely that any association can be made between average-case complexity and worst-case complexity via reductions.<ref name="bog06"/>