This book was typeset with TEX using L ATEX and many further formatting packages. The pictures were prepared using pstricks, xfig, gnuplot and gmt. All numerals in this text are recycled. Für meine Eltern Preface Avoid reality at all costs — fortune(6) As the inclined reader will find out soon enough, this thesis is not about deeply involved mathematics as a mean in itself, but about how to apply mathematics to solve real-world problems. We will show how to shape, forge, and yield our tool of choice to rapidly answer questions of concern to people outside the world of mathematics. But there is more to it. Our tool of choice is software. This is not unusual, since it has become standard practice in science to use software as part of experiments and sometimes even for proofs. But in order to call an experiment scientific it must be reproducible. Is this the case?

: VLSI layout applications yield instances of the Steiner tree problem over grid graphs with holes, which are considered hard to be solved by current methods. In particular, preprocessing techniques developed for Steiner problems over general graphs are not likely to reduce significantly such VLSI instances. We propose a new preprocessing procedure, combining earlier ideas from the literature in such a way that they are now effective over VLSI problems. Testing this procedure over the 116 instances available in the SteinLib, we obtained significant reductions within reasonable computational times. These reductions allowed a branch-and-cut to solve 28 out of 32 open instances in the SteinLib, some with more than 20,000 edges. Keywords: Steiner Problem, Pre-processing, VLSI Design. Resumo: O projeto de circuitos VLSI demanda soluc~oes de instancias do problema de Steiner em grafos. Essas instancias tem como caracter'istica a estrutura de grade bi-dimensional com buracos e s~ao consideradas de dif'icil resoluc~ao pelos m'etodos conhecidos. Em particular, as t'ecnicas de pr'e-processamento desenvolvidas pra problemas de Steiner em grafos gerais n~ao funcionam bem. Este trabalho apresenta um novo procedimento de pr'e-processamento, adaptando testes de reduc~ao j'a conhecidos na literatura tornando-os efetivos para as instancias de VLSI. O procedimento foi aplicado sobre 116 instancias de VLSI da SteinLib, dispon'iveis na rede. Foram obtidas reduc~oes bastante significativas, permitindo que um algoritmo de branch-and-cut resolvesse 28 das 32 instancias de VLSI da SteinLib at'e ent~ao em em aberto. Palavras-chave: Problema de Steiner, Pr'e-processamento, Projeto de VLSI. 1

We propose in this work 50 new test instances for the Steiner problem in graphs. These instances are characterized by large integrality gaps (between the optimal integer solution and that of the linear programming relaxation) and symmetry aspects which make them harder to both exact methods and heuristics than the test instances currently in use for the evaluation and comparison of existing and newly developed algorithms. Our computational results indicate that these new instances are not amenable to reductions by current preprocessing techniques and that not only do the linear programming lower bounds show large gaps, but they are also hard to be computed. State-of-the-art heuristics, which found optimal solutions for almost all test instances currently in use, faced much more difficulties for the new instances. Fewer optimal solutions were found and the numerical results are more discriminant, allowing a better assessment of the effectiveness and the relative behavior of different heuristics.

The Steiner tree problem is a classical NP-hard optimization problem with a wide range of practical applications. In an instance of this problem, we are given an undirected graph G = (V,E), a set of terminals R ⊆ V, and non-negative costs ce for all edges e ∈ E. Any tree that contains all terminals is called a Steiner tree; the goal is to find a minimum-cost Steiner tree. The vertices V\R are called Steiner vertices. The best approximation algorithm known for the Steiner tree problem is a greedy algorithm due to Robins and Zelikovsky (SIAM J. Discrete Math, 2005); it achieves a performance guarantee of 1 + ln3 2 ≈ 1.55. The best known linear programming (LP)-based algorithm, on the other hand, is due to Goemans and Bertsimas (Math. Programming, 1993) and achieves an approximation ratio of 2 − 2/|R|. In this paper we establish a link between greedy and LP-based approaches by showing that Robins and Zelikovsky’s algorithm can be viewed as an iterated primal-dual algorithm with respect to a novel LP relaxation. The LP used in the first iteration is stronger than the well-known bidirected cut relaxation. An instance is b-quasi-bipartite if each connected component of G\R has at most b vertices. We show that Robins ’ and Zelikovsky’s algorithm has an approximation ratio better than 1 + ln3 2 for such instances, and we prove that the integrality gap of our LP is between 8 7

Some of the most widely used constructive heuristics for the Steiner Problem in Graphs are based on algorithms for the Minimum Spanning Tree problem. In this paper, we examine efficient implementations of heuristics based on the classic algorithms by Prim, Kruskal, and Boruvka. An extensive experimental study indicates that the theoretical worst-case complexity of the algorithms give little information about their behavior in practice. Careful implementation improves average computation times not only significantly, but asymptotically. Running times for our implementations are within a small constant factor from that of Prim's algorithm for the Minimum Spanning Tree problem, suggesting that there is little room for improvement.

We extend a well known uncrossing technique in linear programs (LPs) to work with partitions. Using this technique, we tie together three previously unrelated papers on Steiner trees, by showing that the following three values are equal: (1) the objective value of a subtour based LP by Polzin and Vahdati Daneshmand; (2) the objective value of a partition based LP by Könemann and Tan; (3) a “maximum gainless tree ” quantity used by Karpinski and Zelikovsky. These LPs are known to be stronger than the bidirected cut relaxation; we conjecture that in preprocessed graphs, these LPs are exactly as strong as the bidirected cut relaxation, which would add a surprising fourth item to our list. 1

I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii The Steiner tree problem is a classical, well-studied, NP-hard optimization problem. Here we are given an undirected graph G =(V,E), a subset R of V of terminals, and nonnegative costs ce for all edges e in E. A feasible Steiner tree for a given instance is a tree T in G that spans all terminals in R. The goal is to compute a feasible Steiner tree of smallest cost. In this thesis we will focus on approximation algorithms for this problem: a c-approximation algorithm is an algorithm that returns a tree of cost at most c times that of an optimum solution for any given input instance. In a series of papers throughout the last decade, the approximation guarantee c for the Steiner tree problem has been improved to the currently best known value of 1.55 [39]. Robins ’ and Zelikovsky’s algorithm as well as most of its predecessors are greedy

Computing Steiner Minimum Trees in Hamming Metric Computing Steiner minimum trees in Hamming metric is a well studied problem that has applications in several fields of science such as computational linguistics and computational biology. Among all methods for finding such trees, algorithms using variations of a branch and bound method developed by Penny and Hendy have been the fastest for more than 20 years. In this paper we describe a new pruning approach that is superior to previous methods and its implementation. 1

Abstract. We investigate hypergraphic LP relaxations for the Steiner tree problem, primarily the partition LP relaxation introduced by Könemann et al. [Math. Programming, 2009]. Specifically, we are interested in proving upper bounds on the integrality gap of this LP, and studying its relation to other linear relaxations. Our results are the following. Structural results: We extend the technique of uncrossing, usually applied to families of sets, to families of partitions. As a consequence we show that any basic feasible solution to the partition LP formulation has sparse support. Although the number of variables could be exponential, the number of positive variables is at most the number of terminals. Relations with other relaxations: We show the equivalence of the partition LP relaxation with other known hypergraphic relaxations. We also show that these hypergraphic relaxations are equivalent to the well studied bidirected cut relaxation, if the instance is quasibipartite. Integrality gap upper bounds: We show an upper bound of √ 3. = 1.729 on the integrality gap of these hypergraph relaxations in general graphs. In the special case of uniformly quasibipartite instances, we show an improved upper bound of 73/60. = 1.216. By our equivalence theorem, the latter result implies an improved upper bound for the bidirected cut relaxation as well. 1

We present efficient algorithms that implement four local searches for the Steiner problem in graphs: vertex insertion, vertex elimination, key-path exchange, and key-vertex elimination. In each case, we show how to find an improving solution (or prove that none exists in the neighborhood) in O(mlog n) time on graphs with n vertices and m edges. Many of the techniques and data structures we use are relevant in the study of dynamic graphs in general, beyond Steiner trees. Besides the theoretical interest, our results have practical impact: these local searches have been shown to find goodquality solutions in practice, but high running times limited their applicability. 1