No abstract

No abstract

We study output-sensitive algorithms and complexity for multiobjective combinatorial optimization problems. In this computational complexity framework, an algorithm for a general enumeration problem is regarded efficient if it is output-sensitive, that is, its running time is bounded by a polynomial in the input and the output size. We provide both practical examples of multiobjective combinatorial optimization problems for which such an efficient algorithm exists as well as problems for which no efficient algorithm exists under mild complexity theoretic assumptions. KEYWORDScombinatorial optimization, linear programming, multiobjective optimization, output-sensitive complexity 1 1.1 gives us problems, which are not harder than a variant of the problem where we also want to find solutions. Hence, proving hardness of the problem as defined in Definition 1.1 will lead to a hardness result for the problem including finding of a representative solution.

German nuclear power phase out in 2022 leads to significant reconstruction of the energy transmission system. Thus, efficient identification of practical transmission routes with minimum impact on ecological and economical interests is of growing importance. Due to the sensitivity of Germany's public to grid expansion (especially in case of overhead lines), the participation and planning process needs to provide a high degree of openness and accountability. Therefore, a new methodological approach for the computer-assisted finding of optimal power-line routes considering planning, ecological and economic decision criteria is presented. The approach is implemented in a tool-chain for the determination of transmission line routes (and sets of transmission line route alternatives) based on multi-criteria optimisation. Additionally, a decision support system, based on common Geographic Information Systems (GIS), consisting of interactive visualisation and exploration of the solution space is proposed.

We consider the multiobjective shortest path (MOSP) problem. While known approximation algorithms allow a polynomial running time, the degrees of these polynomials are dependent on the number of objective functions. Unfortunately, this also holds true for their best-case. Exact algorithms, while attaining an exponential worst-case running time even in the number of nodes, allow for far better best-case performance and are thus preferred in practice. We introduce a new general approximation framework for MOSP. It aims at combining strong worst-case guarantees with practically useful performance. It allows for various labeling strategies as employed by exact algorithms; thus, decades of research can be utilized. We conduct a comprehensive computational study to compare our framework to known approximations and exact algorithms. For many, this is their first practical investigation. Furthermore, this is the first time that graphs of practically relevant sizes as well as real-world instances are considered in the context of MOSP approximation. The results show that our framework is superior to the known approximation methods in running time and quality. They also demonstrate the usefulness and limits of approximations compared to exact methods.