Structural solutions to maximum independent set and related problems

In this thesis, we study some fundamental problems in algorithmic graph theory. Most natural problems in this area are hard from a computational point of view. However, many applications demand that we do solve such problems, even if they are intractable. There are a number of methods in which we can try to do this: 1) We may use an approximation algorithm if we do not necessarily require the best possible solution to a problem. 2) Heuristics can be applied and work well enough to be useful for many applications. 3) We can construct randomised algorithms for which the probability of failure is very small. 4) We may parameterize the problem in some way which limits its complexity. In other cases, we may also have some information about the structure of the instances of the problem we are trying to solve. If we are lucky, we may and that we can exploit this extra structure to find efficient ways to solve our problem. The question which arises is - How far must we restrict the structure of our graph to be able to solve our problem efficiently? In this thesis we study a number of problems, such as Maximum Indepen- dent Set, Maximum Induced Matching, Stable-II, Efficient Edge Domina- tion, Vertex Colouring and Dynamic Edge-Choosability. We try to solve problems on various hereditary classes of graphs and analyse the complexity of the resulting problem, both from a classical and parameterized point of view

Approximation Schemes for Machine Scheduling

In the classical problem of makespan minimization on identical parallel machines, or machine scheduling for short, a set of jobs has to be assigned to a set of machines. The jobs have a processing time and the goal is to minimize the latest finishing time of the jobs. Machine scheduling is well known to be NP-hard and thus there is no polynomial time algorithm for this problem that is guaranteed to find an optimal solution unless P=NP. There is, however, a polynomial time approximation scheme (PTAS) for machine scheduling, that is, a family of approximation algorithms with ratios arbitrarily close to one. Whether a problem admits an approximation scheme or not is a fundamental question in approximation theory. In the present work, we consider this question for several variants of machine scheduling. We study the problem where the machines are partitioned into a constant number of types and the processing time of the jobs is also dependent on the machine type. We present so called efficient PTAS (EPTAS) results for this problem and variants thereof. We show that certain cases of machine scheduling with assignment restrictions do not admit a PTAS unless P=NP. Moreover, we introduce a graph framework based on the restrictions of the jobs and use it in the design of approximation schemes for other variants. We introduce an enhanced integer programming formulation for assignment problems, show that it can be efficiently solved, and use it in the EPTAS design for variants of machine scheduling with setup times. For one of the problems, we show that there is also a PTAS in the case with uniform machines, where machines have speeds influencing the processing times of the jobs. We consider cases in which each job requires a certain amount of a shared renewable resource and the processing time is depended on the amount of resource it receives or not. We present so called asymptotic fully polynomial time approximation schemes (AFPTAS) for the problems