Average-case Complexity

We review the many open questions and the few things that are known about the average-case complexity of computational problems. We shall follow the presentations of Impagliazzo, of Goldreich, and of Bogdanov and the author, and focus on the following subjects. (i). Average-case tractability. What does it mean for a problem to have an "efficient on average'' algorithm with respect to a distribution of instances? There is more than one ``correct'' answer to this question, and a numberof subtleties arise, which are interesting to discuss. (ii) Worst case versus average-case. Is the existence of hard-on-averageproblems in a complexity class equivalent to the existence of worst-case-hardproblems? This is the case for complexity classes like PSPACE and EXP, but it is openfor NP, with partial evidence pointing to a negative answer. (To be sure, we believethat hard-on-average, and also worst-case hard problems, exist in NP, and if so theirexistence is ``equivalent'' in the way two true statements are logically equivalent. There is, however, partial evidence that such an equivalence cannot be establishedvia reductions. It is also known that such an equivalence cannot be established viaany relativizing technique.) (iii) Amplification of average-case hardness. A weak sense in which aproblem may be hard-on-average is that every efficient algorithm fails on a noticeable(at least inverse polynomial) fraction of inputs; a strong sense is that noalgorithm can do much better than guess the answer at random. In many settings,the existence of problems of weak average-case complexity implies the existenceof problems, in the same complexity class, of strong average-case complexity.It remains open to prove such equivalence in the setting of uniform algorithmsfor problems in NP. (Some partial results are known even in this setting.) (iv) Reductions and Completeness. Levin initiated a theoryof completeness for distributional problems under reductions that preserveaverage-case tractability. Even establishing the existence of an NP-completeproblem in this theory is a non-trivial (and interesting) result.