Average case approximability of optimisation problems

This thesis combines average-case complexity theory with the approximability of optimisation problems. Both are ways of dealing with the fact that many computational problems are not solvable in polynomial time, unless P = NP. A theoretical framework is established that allows both the classification of optimisation problems with respect to their average-case approximability and the study of the structural properties of the resulting average-case approximation classes. Instead of giving worst-case bounds on the time required to decide membership for a decision problem, average-case complexity focuses on the average time that is needed for this task. The average time is taken with respect to a probability distribution on the instances, so average-case properties are not given for a decision problem alone but for a decision problem combined with an input distribution: a distributional problem. For distributional optimisation problems, not only approximation algorithms that run in average polynomial time rather than worst-case polynomial time are of interest. When searching for approximation algorithms for an optimisation problem one aims for algorithms with small performance ratio. In order to relax the worst-case requirements on the performance ratio to average-case requirements, notions are needed to express approximability within a factor that is constant, polynomial or exponential on average. While respective concepts exist for the latter two, the notion of functions that are constant on average is introduced in this thesis. For a number of results on the average behaviour of approximation algorithms for practical problems it is shown how they fit into the new framework. With the framework of definitions established, we can examine the structural properties of the average-case approximation classes. Introducing a reduction that preserves average-case approximability, it is shown that the class of NP-optimisation problems with P-computable input distributions has complete problems. Differences between the worst-case and the average-case setting become apparent when looking at the average-case variant of the worst-case relation “P = NP ⇔ every NP-optimisation problem is solvable in polynomial time”. The question whether NP is easy on average – which means that all NP-problems