Hybrid Metaheuristics for Generalized Network Design Problems

In this thesis, we consider several generalized network design problems (NDPs) which belong to the family of NP-hard combinatorial optimization problems. In contrast to classical NDPs, the generalized versions are defined on graphs whose node sets are partitioned into clusters. The goal is to find a subgraph which spans exactly one node from each cluster and also meets further constraints respectively. Applicable methodologies for solving combinatorial optimization problems can roughly be divided into two mainstreams. The first class consists of algorithms which aim to solve these problems to proven optimality – provided that they are given enough run-time and memory. This thesis starts with a brief introduction to linear and integer linear programming techniques since popular algorithms like branch-andbound, branch-and-cut, etc. are based on them. The second class are metaheuristics which compute approximate solutions but usually require significantly less runtime. By combining these two classes, we are able to form collaboration approaches that benefit from advantages of both sides. We will examine various possibilities of such combinations and some of them will be used to solve the NDPs in this thesis. The first considered NDP is the generalized minimum spanning tree problem. Given a graph whose nodes are partitioned into clusters, we seek a minimum spanning tree which connects exactly one node from each cluster. A variable neighborhood search (VNS) approach will be presented that uses three different neighborhood types. Two of them work in complementary ways in order to maximize search performance. Both are large in the sense that they contain exponentially many candidate solutions, but efficient polynomial-time algorithms are used to identify best neighbors. For the third neighborhood type we apply integer programming to optimize local parts within candidate solution trees.