Date Dr Hab. Šukasz Kowalik Capacitated Dominating Set, Minimum Maximal Irredundant Set Acm Classication: F.2.2, G.2.1, G.2.2. 4 Chapter 1

All NP-complete problems can be solved in exponential time by enumerating the space of possible solutions. In the area of moderatelyexponential algorithms, we seek for exact algorithms for NP-complete problems that are faster than the naive ones. Surprisingly, often such results exist and their development leads to a good insight into a considered problem. In the rst part of this dissertation we give algorithms for three problems: Capacitated Dominating Set,Minimum Maximal Irredundant Set and one job scheduling problem. In all cases there exists a simple O(2)-time algorithm, and our results break the natural 2-barrier. All three results are based on new observations on the considered problems that allow us to limit the search space. In the parameterized complexity setting we assume that a given instance I (of a usually NP-complete problem) comes up with a parameter k. We seek for algorithms (called xed-parameter algorithms) working in time f(k)|I| for some computable function f . In other words, we want the superpolynomial factor in the time complexity, probably unavoidable for NP-complete problems, to depend on the parameter only. A special case of xed-parameter algorithms is the eld of kernelization, where one seeks for a polynomialtime preprocessing algorithms that shrink a given instance to one with size bounded by a function of the parameter k. The second part of this dissertation is devoted to two results in this elds. First, we show the rst known xed-parameter algorithm for Subset Feedback Vertex Set, a problem closely related to graph-cutting problems that are now central in parameterized complexity. Second, we prove that, unless the polynomial hierarchy collapses up to its third level, many problems involving connectivity requirement do not admit a kernelization algorithm with polynomial (in the parameter) guarantee on the output size, even if the input graph is restricted to be of bounded degeneracy. This proves that known kernelization algorihtms for graphs excluding a xed minor (being a subclass of graphs of bounded degeneracy) indeed require the topological properties of these graph classes.