Precise average case complexity

Abstract

A new definition is given for the average growth of a function f: Σ* → IN with respect to a probability measure μ on Σ*. This allows us to define meaningful average case distributional complexity classes for arbitrary time bounds (previously, one could only distinguish between polynomial and superpolynomial growth). It is shown that basically only the ranking of the inputs by decreasing probabilities are of importance. To compare the average and worst case complexity of problems we study average case complexity classes defined by a time bound and a bound on the complexity of possible distributions. Here, the complexity is measured by the time to compute the rank functions of the distributions. We obtain tight and optimal separation results between these average case classes. Also the worst case classes can be embedded into this hierarchy. They are shown to be identical to average case classes with respect to distributions of exponential complexity. These ideas are finally applied to study the average case complexity of problems in NP. A reduction between distributional problems is defined for this new approach. We study the average case complexity class AvP consisting of those problems that can be solved by DTMs on the average in polynomial time for all distributions with efficiently computable rank function. Fast algorithms are known for some NP-complete problems under very simple distributions. For langugages in NP we consider the maximal allowable complexity of distributions such that the problem can still be solved efficiently by a DTM, at least on the average. As an example we can show that either the satisfiability problem remains hard, even for simple distributions, or NP is contained in AvV, that means every problem in NP can be solved efficiently on the average for arbitrary not too complex distributions.