Solving Two Generalized Network Design Problems with Exact and Heuristic Methods

This thesis considers two NP hard generalized network design problems, where the nodes of a graph are partitioned into clusters and exactly one node from each cluster must be connected. The objective of both problems is to identify for a given graph a subgraph with minimal total edge costs and satisfying certain constraints. The Generalized Minimum Spanning Tree (GMST) problem extends the classical Minimum Spanning Tree problem by seeking a connected cycle-free subgraph containing exactly one node from every cluster. This problem is solved by a Variable Neighborhood Search (VNS) approach which uses three different neighborhood types. The first one is focused on the nodes selected within a concrete solution, while the second one first selects the global edges between the clusters. Both are large in the sense that they contain exponentially many candidate solution, but efficient polynomial-time algorithms are used to identify best neighbors. The third neighborhood type uses Integer Linear Programming (ILP) to solve parts of the problem to provable optimality. Tests on Euclidean and random instances with up to 1280 nodes indicate especially on instances with many nodes per cluster significant advantages over previously published metaheuristic approaches. Extending the classical Minimum Edge Biconnected Network problem, goal of the Generalized Minimum Edge Biconnected Network (GMEBCN) problem is to obtain a subgraph connecting exactly one node from each cluster and containing no bridges. Two different Variable Neighborhood Search (VNS) approaches using four different neighborhood types, are presented for this problem. The first one focuses on optimizing the used nodes, while the second one puts more emphasis on the arrangement of them. Two different versions exists for both neighborhoods. The simpler ones operate in a straightforward way on the nodes of the solution-graph while the more sophisticated versions consider the so-called “reduced graph”. Using this significant smaller graph, it is easy to determine the best used nodes for the majority of clusters in an optimal way. The third neighborhood type optimizes a solution by first adding a new edge and then removing as many unnecessary edges as possible. Finally the last neighborhood optimizes both the used nodes as well as the edges. It is based on changing the used nodes within exactly one cluster and removing all incident edges which divides the graph into several edge-biconnected components. Afterwards the solution is heuristically augmented until the edge biconnectivity property holds again. Comparing these two approaches on Euclidean and random instances with up to 1280 nodes indicate that the second approach, using the more sophisticated neighborhoods and having higher computational complexity, is able to outperform the simpler, but much faster one significantly with respect to solution quality.