Exploring Parameter Spaces in Coping with Computational Intractability

Abstract

This thesis is about systematic approaches to identify tractable cases of computationally hard problems. It studies the influence of specific parameters on the computational complexity of the problems. Hence, parameterized complexity provides a natural and fruitful framework which forms the theoretical basis of our investigations. In a nutshell, parameterized algorithmics deals with identifying parameters in computationally hard problems (mostly, so-called NP-hard problems) and exploits them in order to design parameterized algorithms whose (presumably) unavoidable exponential running time part solely depends on a parameter value. This approach is strongly motivated by the often observed phenomenon that, when dealing with NP-hard problems in practical applications, so-called heuristic algorithms which are tuned to characteristics of the input data are well-performing, both in terms of running time as well as solution quality. The most plausible explanation for this behavior is that problem instances emerging from practical applications are not worst-case instances but rather admit certain, possibly application-specific, structures. These structures are often far from being obvious or easy to identify, and in many cases empirical studies of typical data sets help to discover them. Probably one of the most prominent examples for this is the small world phenomenon roughly stating that in social networks, although each node has rather few neighbors, the length of a shortest path between nodes is often at most ten. Exploiting small parameter values that are implied by these structures for the development of exact and “efficient” solving algorithms is the central concern of parameterized algorithmics. In this thesis we describe three approaches to identify structures which determine the computational complexity of a problem. These three approaches naturally lead to so-called parameter spaces, that is, several parameters that are related to each other in some way which may allow to transfer tractability and intractability results between them. Hence they pave the way for a systematic and clear analysis of computational complexity and they help to chart the “border of intractability”. We next describe our three approaches to identify tractable cases of NP-hard problems and outline our corresponding case studies. The first approach is tuned for graph problems and suggests to consider the computational complexity of a graph problem, which is NP-hard or