Algorithms and experiments for parameterized approaches to hard graph problems

This thesis is about the design, analysis, implementation, and experimental evaluation of algorithms for hard graph problems. The aim is to establish that the concept of fixed-parameter tractability, and in particular novel algorithmic techniques whose development was driven by this concept, can lead to practically useful programs for exactly solving real-world problem instances. In particular, we examine the three graph problems Clique Cover, Balanced Subgraph, and Minimum-Weight Path. In the Clique Cover problem, the task is to find a minimum-cardinality set of cliques that covers all edges. It has applications in compiler optimization, computational geometry, and applied statistics. The Balanced Subgraph problem asks for a 2-coloring of a graph that minimizes the inconsistencies with given edge labels. It has applications in social networks, systems biology, and integrated circuit design. The Minimum-Weight Path problem is to find a minimum-weight simple path of a given length. It has applications in systems biology and text mining. All three problems, as many other practically relevant graph problems, are NP-hard, implying that presumably any exact algorithm requires time exponential in the input size. The idea of parameterized complexity is to take a two-dimensional view, where in addition to the input size we consider a parameter k; a typical parameter is the size of the solution. A problem is called fixed-parameter tractable if an instance of size n can be solved in f(k) · nO(1) time, where f is an arbitrary computable function that absorbs the exponential part of the running time. Thus, whenever the parameter turns out to be small in practice, we can expect good running times. Various techniques to design fixed-parameter algorithms have been suggested. We examine three of them: data reduction, iterative compression, and colorcoding. Data reduction. Data reduction is a classic way of dealing with hard problems: before starting the actual solving process, one tries to reduce the size of the instance by removing or simplifying parts, for example because we can immediately infer that they are irrelevant to the solution. We present efficient and effective data reduction rules for Clique Cover, which in particular allow to prove that Clique Cover is fixed-parameter tractable. Combined with a simple search tree algorithm, this allows for exact problem solutions in competitive time. This is confirmed by experiments with real-world and synthetic data. We then present a novel data reduction for Balanced Subgraph based on finding small separators and a novel gadget construction scheme. The data reduction scheme can be applied to a large number of graph problems where a coloring or a subset of the vertices is sought. Our implementation, which