Parameterized NP-hard problems are NP-hard problems that are associated with special variables called parameters. One example of the problem is to find simple paths of length k in a graph, where the integer k is the parameter. We call this problem the p-path problem. The p-path problem is the parameterized version of the well-known NP-complete problem - the longest simple path problem. There are two main reasons why we study parameterized NP-hard problems. First, many application problems are naturally associated with certain parameters. Hence we need to solve these parameterized NP-hard problems. Second, if parameters take only small values, we can take advantage of these parameters to design very effective algorithms. If a parameterized NP-hard problem can be solved by an algorithm of running time in form of f(k)nO(1), where k is the parameter, f(k) is independent of n, and n is the input size of the problem instance, we say that this parameterized NP-hard problem is fixed parameter tractable (FPT). If a problem is FPT and the parameter takes only small values, the problem can be solved efficiently (it can be solved almost in polynomial time). In this dissertation, first, we introduce several techniques that can be used to design efficient algorithms for parameterized NP-hard problems. These techniques include branch and bound, divide and conquer, color coding and dynamic programming, iterative compression, iterative expansion and kernelization. Then we present our results about how to use these techniques to solve parameterized NP-hard problems, such as the p-path problem and the pd-feedback vertex set problem. Especially, we designed the first algorithm of running time in form of f(k)nO(1) for the pd-feedback vertex set problem. Thus solved an outstanding open problem, i.e. if the pd-feedback vertex set problem is FPT. Finally, we will introduce how to use parameterized algorithm techniques to solve the signaling pathway problem and the motif finding problem from bioinformatics.

According to the theory of NPcompleteness, many problems that have important realworld applications are NPhard. This excludes the possibility of solving them in polynomial time unless P=NP. A number of approaches have been proposed in dealing with NPhard problems, among them are approximation algorithms and parameterized algorithms. The study of approximation algorithms tries to find good enough solutions instead of optimal solutions in polynomial time, while parameterized algorithms try to give exact solutions when a natural parameter is small. In this thesis, we study the structural properties of parameterized computation and approximation algorithms for NP optimization problems. In particular, we investigate the relationship between parameterized complexity and polynomialtime approximation scheme (PTAS) for NP optimization problems. We give nice characterizations for two important subclasses in PTAS: Fully Polynomial Time Approximation Scheme (FPTAS) and Effcient Polynomial Time Approximation Scheme (EPTAS), using the theory of parameterized complexity. Our characterization of the class FPTAS has its advantages over the former characterizations, and our characterization of EPTAS is the first systematic investigation of this new but important approximation class. We develop new techniques to derive strong computational lower bounds for certain parameterized problems based on the theory of parameterized complexity. For example, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the clique problem could not be solved in time O(f (k)no(k)) for any function f . This lower bound matches the upper bound of the trivial algorithm that simply enumerates and checks all subsets of k vertices in the given graph of n vertices. We then extend our techniques to derive computational lower bounds for PTAS and EPTAS algorithms of NP optimization problems. We prove that certain NP optimization problems with known PTAS algorithms have no PTAS algorithms of running time O(f (1/Epsilon)no(1/Epsilon)) for any function f . Therefore, for these NP optimization problems, although theoretically they can be approximated in polynomial time to an arbitrarily small error bound Epsilon, they have no practically effective approximation algorithms for small error bound Epsilon. To our knowledge, this is the first time such lower bound results have been derived for PTAS algorithms. This seems to open a new direction for the study of computational lower bounds on the approximability of NP optimization problems.

The Master’s thesis deals with production scheduling in an industrial company. It uses the means of artificial intelligence to develop an appropriate production schedule in a generalized Flow-shop Programming problem. This problem can be solved by application which is a result of this thesis and was prepaired with use of the software Matlab 7.1 and its Genetic Algorithm and Direct Search toolbox. There is a part devoted to the use of advanced production systems (APS) and the concept of the operative production planning in praxis as well. The thesis pays attention to various optimization models in production scheduling and supply chain management too.