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Author
Date
2019Type
- Doctoral Thesis
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yes
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Abstract
In this dissertation, we discuss several topics in the context of mixed integer linear programming in variable dimension, both from a theoretical and a practical point of view.
Lying in the intersection of Mathematics and Computer Science, this problem has rich theory and a wide variety of applications. However, many questions in integer linear programming are computationally intractable or known to be NP-hard.
Much of the research in integer linear programming is driven by the goal of utilizing positive complexity results from fixed dimensional problems to related problems in variable dimension.
In the first part of the thesis, we study a reformulation of integer linear programs by means of mixed integer linear programs with fewer integer variables. Our reformulations capture the underlying algebraic and geometric structure of the problem. Our modeling decision is influenced by the ability to solve mixed integer linear programs efficiently when only few integer variables are present. This leads to mixed integer formulations that can express the set of feasible solutions more elegantly than solely linear inequality descriptions.
Many real-world problems that are modeled by integer linear programs have a combinatorial flavor where a feasible solution can be constructed by selecting elements from a given ground set.
In the second part of the thesis, we study such classes of integer linear optimization problems under a time aspect. For some applications, resources to construct a final feasible solution are not available immediately, but the solution can only be constructed step by step in an incremental fashion. Our target is to find a feasible solution for the global optimization problem that is also reasonably good in the intermediate time steps. In order to get such solutions we construct objective functions that punish the violation of the constraints of the global problem in intermediate steps. We show that many of the problems studied inherit the NP-hardness from the related global problem. On the other hand, polynomially solvable cases are identified.
Driven by the application of base station placement for drone supervision, in a third part of the thesis algorithms and complexity results for a geometric covering problem are studied. Techniques from different areas of mathematical optimization and theoretical computer science are combined to solve a new version of a well-known terrain guarding problem. Show more
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https://doi.org/10.3929/ethz-b-000399560Publication status
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Publisher
ETH ZurichOrganisational unit
03873 - Weismantel, Robert / Weismantel, Robert
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ETH Bibliography
yes
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