SYSTEMATIC KERNELIZATION IN FPT ALGORITHM DESIGN

Data reduction is a preprocessing technique which makes huge and seemingly intractable instances of a problem small and tractable. This technique is often acknowledged as one of the most powerful methods to cope with the intractability of certain NP-complete problems. Heuristics for reducing data can often be seen as reduction rules, if considered from a parameterized complexity viewpoint. Using reduction rules to transform the instances of a parameterized problem into equivalent instances, with size bounded by a function of the parameter, is known as kernelization. This thesis introduces and develops an approach to designing FPT algorithms based on effective kernelizations. This method is called the method of extremal structure. The method operates following a paradigm presented by two lemmas, the kernelization and boundary lemmas. The boundary lemma is used to find theoretical bounds on the maximum size of an instance reduced under an adequate set of reduction rules. The kernelization lemma is invoked to decide the instances which are larger than for some function depending only on the parameter . The first aim of the method of extremal structure is to provide a systematic way to discover reduction rules for fixed-parameter tractable problems. The second is to devise an analytical way to find theoretical bounds for the size of kernels for those problems. These two aims are achieved with the aid of combinatorial extremal arguments. Furthermore, this thesis shows how the method of extremal structure can be applied to effectively solve several NP-complete problems, namely MAX CUT, MAX LEAF SPANNING TREE, NONBLOCKER, -STAR PACKING, EDGE-DISJOINT TRIANGLE PACKING, MAX INTERNAL SPANNING TREE and MINIMUM MAXIMAL MATCHING.