## 05C05 Trees

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#### Keywords

- Adjazenz-Beziehungen (1)
- Baum <Mathematik> (1)
- Cut (1)
- Graph Theory (1)
- Graphentheorie (1)
- Heuristic (1)
- Heuristik (1)
- Knapsack (1)
- Kombinatorische Optimierung (1)
- Minimal spannender Baum (1)

- Weight-Constrained Minimum Spanning Tree Problem (2007)
- In an undirected graph G we associate costs and weights to each edge. The weight-constrained minimum spanning tree problem is to find a spanning tree of total edge weight at most a given value W and minimum total costs under this restriction. In this thesis a literature overview on this NP-hard problem, theoretical properties concerning the convex hull and the Lagrangian relaxation are given. We present also some in- and exclusion-test for this problem. We apply a ranking algorithm and the method of approximation through decomposition to our problem and design also a new branch and bound scheme. The numerical results show that this new solution approach performs better than the existing algorithms.

- Minimum Fundamental Cut Basis Problem (2004)
- Any tree in an undirected graph defines a fundamental cut basis. The minimum fundamental cut basis problem is to find a tree minimizing the weight of the corresponding basis. This problem is NP, in the thesis heuristics, relaxations, and numerical results are presented.