Approximation algorithms for scheduling and two-dimensional packing problems

This dissertation thesis is concerned with two topics of combinatorial optimization : scheduling and geometrical packing problems. Scheduling deals with the assignment of jobs to machines in a ‘good’ way, for suitable notions of good. Two particular problems are studied in depth : on the one hand, we consider the impact of machine failure on online scheduling, i.e. what are the consequences of the fact that in real life, machines do not work flawlessly around the clock, but need maintenance intervals or can break down? How do we need to adapt our algorithms to still obtain good overall schedules, and in what settings do we even have a chance to succeed? Our second problem is of a more static nature : in some settings, not every job is permitted on all the machines. A classical example would be that of workers which needs special qualification to execute some jobs, or a certain minimum requirement of memory size of computers, etc. The problem in general is notoriously hard to tackle; we present improved approximation ratios for several special cases. In particular, we derive a polynomial-time approximation scheme for nested interval restrictions, which occur naturally in many practical applications. Our final topic is two-dimensional geometric bin packing, the problem of packing rectangular objects into the minimum number of containers of identical size (figuratively speaking, we are arranging advertisements of fixed dimensions into the minimum number of print pages). It is known that no approximation ratio better than 2 is possible for this problem, unless P = NP; we present an algorithm that guarantees this ratio.