Phase transitions and typical-case complexity: Easy (hard) aspects of hard (easy) problems.

摘要

In this thesis, we study theoretically and empirically the typical-case hardness of randomly-generated instances of several algorithmic problems that are of interest in artificial intelligence research. For randomly-generated instances of constraint satisfaction problems (CSP), we identified a new class of algorithmically exploitable structures and proved that under certain instance distributions, random instances contain such structures with high probability (Chapter 4). In an effort to find a way to eliminate these structures from randomly-generated CSP instances, we established an interesting connection between the notion of constraint consistency in the literature and the resolution complexity of random CSP instances. By embedding a recursive structure called consistency core into random CSP models, we proposed a novel scheme to generate random CSP instances with theoretically guaranteed resolution complexity and empirically confirmed hardness (Chapter 5). Our proposal resolved the long-standing problem of generating hard random CSP instances with bounded domain size that has troubled the society for several years.; While all of the results in Chapters 4 and 5 are aimed at backtracking search algorithms, we investigated in Chapter 6 the typical-case behavior of random instances in terms of the dynamic programming algorithms whose time and space complexities are exponential in the treewidth of the underlying structures. This type of algorithm has been widely used in the study of Bayesian network inference and CSPs. We established an improved lower bound on the threshold for a random graph to have a treewidth linear in the graph size. Similar techniques were then applied to random CSPs, random Bayesian networks, and fitness landscape models in computational biology and evolutionary computation.