Backdoors To Typical Case Complexity

Abstract

There has been significant recent progress in reasoning and constraint processing methods. In areas such as planning and finite model-checking, current solution techniques can handle combinatorial problems with up to a million variables and five million constraints. The good scaling behavior of these methods appears to defy what one would expect based on a worst-case complexity analysis. In order to bridge this gap between theory and practice, we propose a new framework for studying the complexity of these techniques on practical problem instances. In particular, our approach incorporates general structural properties observed in practical problem instances into the formal complexity analysis. We introduce a notion of "backdoors", which are small sets of variables that capture the overall combinatorics of the problem instance. We provide empirical results showing the existence of such backdoors in real-world problems. We then present a series of complexity results that explain the good scaling behavior of current reasoning and constraint methods observed on practical problem instances.