Approximation algorithms for packing and scheduling problems
Algorithms for solving optimization problems play a major role in the industry. For example in the logistics industry, route plans have to be optimized according to various criteria. However, many natural optimization problems are hard to solve. That is, for many optimization problems no algorithms with running time polynomial in the size of the instance are known. Furthermore, it is a widely accepted assumption that many optimization problems do not allow algorithms that solve the problem optimally in polynomial time. One way of overcoming this dilemma is using approximation algorithms. These algorithms have a polynomial running time, but their solutions are in general not optimal but rather close to an optimum. The main subject of this thesis is approximation algorithms for packing and scheduling problems: For the three-dimensional orthogonal knapsack problem (OKP-3) without rotations we present algorithms with approximation ratios arbitrarily close to 9, 8 and 7. For OKP-3 with 90 degree rotations around the z-axis or around all axes, we present algorithms with approximation ratios arbitrarily close to 6 and 5, respectively. Both for the malleable and for the non-malleable case of the non-preemptive parallel job scheduling problem in which the number of available machines is polynomially bounded in the number of jobs, we present polynomial time approximation schemes. For the cases in which additionally the machines allotted to each job have to be contiguous, we show the existence of approximation algorithms with ratio arbitrarily close to 1.5.
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