On Average Case Complexity of Problems that are Intractable in the Worst Case

The worst case setting seems to be the most important setting for studying complexity of approximately solved problems. Unfortunately, some problems of practical importance are intractable or even unsolvable in the worst case setting and to cope with the inherent difficulty of such problems one has to switch to other settings. A natural alternative is provided by the average case setting under which worst case intractable and/or unsolvable problems become tractable. In this paper we will discuss some results on multivariate integration and function approximation and show by how much the complexity is reduced when the average case is utilized.