A Personal View of Average-Case Complexity

The structural theory of average-case complexity, introduced by Levin, gives a formal setting for discussing the types of inputs for which a problem is difficult. This is vital to understanding both when a seemingly difficult (e.g. NP-complete) problem is actually easy on almost all instances, and to determining which problems might be suitable for applications requiring hard problems, such as cryptography. This paper attempts to summarize the state of knowledge in this area, including some "folklore" results that have not explicitly appeared in print. We also try to standardize and unify definitions. Finally, we indicate what we feel are interesting research directions. We hope that this paper will motivate more research in this area and provide an introduction to the area for people new to it.