## 90C11 Mixed integer programming

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#### Keywords

- Connectivity (1)
- Cycle Decomposition (1)
- Graph Theory (1)
- Mixed Connectivity (1)
- Multileaf collimator (1)
- Multiset Multicover (1)
- Robust Optimization (1)
- large scale integer programming (1)

Dealing with uncertain structures or data has lately been getting much attention in discrete optimization. This thesis addresses two different areas in discrete optimization: Connectivity and covering.
When discussing uncertain structures in networks it is often of interest to determine how many vertices or edges may fail in order for the network to stay connected.
Connectivity is a broad, well studied topic in graph theory. One of the most important results in this area is Menger's Theorem which states that the minimum number of vertices needed to separate two non-adjacent vertices equals the maximum number of internally vertex-disjoint paths between these vertices. Here, we discuss mixed forms of connectivity in which both vertices and edges are removed from a graph at the same time. The Beineke Harary Conjecture states that for any two distinct vertices that can be separated with k vertices and l edges but not with k-1 vertices and l edges or k vertices and l-1 edges there exist k+l edge-disjoint paths between them of which k+1 are internally vertex-disjoint. In contrast to Menger's Theorem, the existence of the paths is not sufficient for the connectivity statement to hold. Our main contribution is the proof of the Beineke Harary Conjecture for the case that l equals 2.
We also consider different problems from the area of facility location and covering. We regard problems in which we are given sets of locations and regions, where each region has an assigned number of clients. We are now looking for an allocation of suppliers into the locations, such that each client is served by some supplier. The notable difference to other covering problems is that we assume that each supplier may only serve a fixed number of clients which is not part of the input. We discuss the complexity and solution approaches of three such problems which vary in the way the clients are assigned to the suppliers.

In this thesis we have discussed the problem of decomposing an integer matrix \(A\) into a weighted sum \(A=\sum_{k \in {\mathcal K}} \alpha_k Y^k\) of 0-1 matrices with the strict consecutive ones property. We have developed algorithms to find decompositions which minimize the decomposition time \(\sum_{k \in {\mathcal K}} \alpha_k\) and the decomposition cardinality \(|\{ k \in {\mathcal K}: \alpha_k > 0\}|\). In the absence of additional constraints on the 0-1 matrices \(Y^k\) we have given an algorithm that finds the minimal decomposition time in \({\mathcal O}(NM)\) time. For the case that the matrices \(Y^k\) are restricted to shape matrices -- a restriction which is important in the application of our results in radiotherapy -- we have given an \({\mathcal O}(NM^2)\) algorithm. This is achieved by solving an integer programming formulation of the problem by a very efficient combinatorial algorithm. In addition, we have shown that the problem of minimizing decomposition cardinality is strongly NP-hard, even for matrices with one row (and thus for the unconstrained as well as the shape matrix decomposition). Our greedy heuristics are based on the results for the decomposition time problem and produce better results than previously published algorithms.