## Fachbereich Mathematik

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- A Finite Dominating Set Algorithm for a Dynamic Location Problem in the Plane (2014)
- A single facility problem in the plane is considered, where an optimal location has to be identified for each of finitely many time-steps with respect to time-dependent weights and demand points. It is shown that the median objective can be reduced to a special case of the static multifacility median problem such that results from the latter can be used to tackle the dynamic location problem. When using block norms as distance measure between facilities, a Finite Dominating Set (FDS) is derived. For the special case with only two time-steps, the resulting algorithm is analyzed with respect to its worst-case complexity. Due to the relation between dynamic location problems for T time periods and T-facility problems, this algorithm can also be applied to the static 2-facility location problem.

- Bicriteria approach to the optimal location of surveillance cameras (2014)
- We consider the problem of finding efficient locations of surveillance cameras, where we distinguish between two different problems. In the first, the whole area must be monitored and the number of cameras should be as small as possible. In the second, the goal is to maximize the monitored area for a fixed number of cameras. In both of these problems, restrictions on the ability of the cameras, like limited depth of view or range of vision are taken into account. We present solution approaches for these problems and report on results of their implementations applied to an authentic problem. We also consider a bicriteria problem with two objectives: maximizing the monitored area and minimizing the number of cameras, and solve it for our study case.

- Transit Dependent Evacuation Planning for Kathmandu Valley: A Case Study (2014)
- Due to the increasing number of natural or man-made disasters, the application of operations research methods in evacuation planning has seen a rising interest in the research community. From the beginning, evacuation planning has been highly focused on car-based evacuation. Recently, also the evacuation of transit depended evacuees with the help of buses has been considered. In this case study, we apply two such models and solution algorithms to evacuate a core part of the metropolitan capital city Kathmandu of Nepal as a hypothetical endangered region, where a large part of population is transit dependent. We discuss the computational results for evacuation time under a broad range of possible scenarios, and derive planning suggestions for practitioners.

- Sink Location to Find Optimal Shelters in Evacuation Planning (2014)
- The sink location problem is a combination of network flow and location problems: From a given set of nodes in a flow network a minimum cost subset \(W\) has to be selected such that given supplies can be transported to the nodes in \(W\). In contrast to its counterpart, the source location problem which has already been studied in the literature, sinks have, in general, a limited capacity. Sink location has a decisive application in evacuation planning, where the supplies correspond to the number of evacuees and the sinks to emergency shelters. We classify sink location problems according to capacities on shelter nodes, simultaneous or non-simultaneous flows, and single or multiple assignments of evacuee groups to shelters. Resulting combinations are interpreted in the evacuation context and analyzed with respect to their worst-case complexity status. There are several approaches to tackle these problems: Generic solution methods for uncapacitated problems are based on source location and modifications of the network. In the capacitated case, for which source location cannot be applied, we suggest alternative approaches which work in the original network. It turns out that latter class algorithms are superior to the former ones. This is established in numerical tests including random data as well as real world data from the city of Kaiserslautern, Germany.

- On the Generality of the Greedy Algorithm for Solving Matroid Base Problems (2013)
- It is well known that the greedy algorithm solves matroid base problems for all linear cost functions and is, in fact, correct if and only if the underlying combinatorial structure of the problem is a matroid. Moreover, the algorithm can be applied to problems with sum, bottleneck, algebraic sum or \(k\)-sum objective functions.

- The Multi Terminal q-FlowLoc Problem: A Heuristic (2011)
- In this paper the multi terminal q-FlowLoc problem (q-MT-FlowLoc) is introduced. FlowLoc problems combine two well-known modeling tools: (dynamic) network flows and locational analysis. Since the q-MT-FlowLoc problem is NP-hard we give a mixed integer programming formulation and propose a heuristic which obtains a feasible solution by calculating a maximum flow in a special graph H. If this flow is also a minimum cost flow, various versions of the heuristic can be obtained by the use of different cost functions. The quality of this solutions is compared.

- Universal Shortest Paths (2010)
- We introduce the universal shortest path problem (Univ-SPP) which generalizes both - classical and new - shortest path problems. Starting with the definition of the even more general universal combinatorial optimization problem (Univ-COP), we show that a variety of objective functions for general combinatorial problems can be modeled if all feasible solutions have the same cardinality. Since this assumption is, in general, not satisfied when considering shortest paths, we give two alternative definitions for Univ-SPP, one based on a sequence of cardinality contrained subproblems, the other using an auxiliary construction to establish uniform length for all paths between source and sink. Both alternatives are shown to be (strongly) NP-hard and they can be formulated as quadratic integer or mixed integer linear programs. On graphs with specific assumptions on edge costs and path lengths, the second version of Univ-SPP can be solved as classical sum shortest path problem.

- A Note On Inverse Max Flow Problem Under Chebyshev Norm (2009)
- In this paper, we study the inverse maximum flow problem under \(\ell_\infty\)-norm and show that this problem can be solved by finding a maximum capacity path on a modified graph. Moreover, we consider an extension of the problem where we minimize the number of perturbations among all the optimal solutions of Chebyshev norm. This bicriteria version of the inverse maximum flow problem can also be solved in strongly polynomial time by finding a minimum \(s - t\) cut on the modified graph with a new capacity function.

- A new sequential extraction heuristic for optimising the delivery of cancer radiation treatment using multileaf collimators (2008)
- Finding a delivery plan for cancer radiation treatment using multileaf collimators operating in ''step-and-shoot mode'' can be formulated mathematically as a problem of decomposing an integer matrix into a weighted sum of binary matrices having the consecutive-ones property - and sometimes other properties related to the collimator technology. The efficiency of the delivery plan is measured by both the sum of weights in the decomposition, known as the total beam-on time, and the number of different binary matrices appearing in it, referred to as the cardinality, the latter being closely related to the set-up time of the treatment. In practice, the total beam-on time is usually restricted to its minimum possible value, (which is easy to find), and a decomposition that minimises cardinality (subject to this restriction) is sought.

- Minimum Cut Bases in Undirected Networks (2007)
- Given an undirected, connected network G = (V,E) with weights on the edges, the cut basis problem is asking for a maximal number of linear independent cuts such that the sum of the cut weights is minimized. Surprisingly, this problem has not attained as much attention as its graph theoretic counterpart, the cycle basis problem. We consider two versions of the problem, the unconstrained and the fundamental cut basis problem. For the unconstrained case, where the cuts in the basis can be of an arbitrary kind, the problem can be written as a multiterminal network flow problem and is thus solvable in strongly polynomial time. The complexity of this algorithm improves the complexity of the best algorithms for the cycle basis problem, such that it is preferable for cycle basis problems in planar graphs. In contrast, the fundamental cut basis problem, where all cuts in the basis are obtained by deleting an edge, each, from a spanning tree T is shown to be NP-hard. We present heuristics, integer programming formulations and summarize first experiences with numerical tests.