Approximation Algorithms for 2-dimensional Packing and Related Scheduling Problems

Main subject of this thesis are approximation algorithms for scheduling and packing, two classical geometrical problems in combinatorial optimization. It is divided into three parts. In the first part we consider a generalization of the strip packing problem or geometrical cutting stock problem. In strip packing a given set of rectangles has to be placed into a strip of fixed width and infinite height minimizing the total height used. In the generalized problem setup there are N strips available in which the rectangles have to be allocated and the objective is to minimize the maximum height used. The second part deals with a related scheduling problem. Here we are given parallel jobs instead of rectangles that have to be executed in N platforms of processors. Such a job can be identified with a rectangle, but in contrast to rectangle packing in parallel job scheduling we are allowed to cut the jobs into vertical slides and place them into the platforms as long as all slides of a job start at the same time in the same platform. In the third part we investigate scheduling on uniform processors, a fundamental 1-dimensional scheduling problem. Here the jobs are described by a processing time only and have to be assigned to a set of processors that may run at different speeds.