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University of Copenhagen Nesting Problems and Steiner Tree Problems Nielsen, Benny Kjær Publication date: 2008 Document version Publisher's PDF, also known as Version of record Citation for published version (APA): Nielsen, B. K. (2008). Nesting Problems and Steiner Tree Problems. Download date: 06. Oct. 2021 Nesting Problems and Steiner Tree Problems Benny Kjær Nielsen [email protected] DIKU, University of Copenhagen November, 2007 Preface A large part of research is a search through a maze built of existing literature. Numerous journal papers, conference proceedings, technical reports and books are read in order to figure out what has been done and what has (probably) not been done within a certain research field. Occasionally one thinks of a new idea and a new search begins in the relevant part of the maze to answer the question: Did someone have the same or a similar idea? Working on the idea might reveal that it is a bad idea — a blind alley in the maze — or it may just seem like the path goes on forever and eventually one decides to go back. At some point, one reaches a level of understanding of the research field which makes it easier to separate good ideas from bad ideas. In a sense, one rises above the maze. The maze is still there, but it is much easier to find a successful path. This thesis contains two major research fields; nesting problems and Steiner tree problems. In the analogy of the research maze, I managed to rise above the maze for some parts of these research fields and I stayed firmly on the ground for other parts. Many blind or seemingly endless alleys were attempted. One of these seemingly endless (and maybe blind) alleys deserves special mentioning. In cooperation with Marco E. L¨ubbecke1, Martin Zachariasen and Pawel Winter, it was attempted to solve the “rectilinear Steiner tree problem with obstacles” using integer programming and column generation. A major implementation was done by Marco and I which included an increasing number of advanced strategies in order to improve the results. We never reached the end of the alley, but Marco deserves my thanks for his great efforts, especially since I never had the pleasure of co-authoring any other papers with him. Also, a special thank you to each of my co-authors in this thesis. They are, in alphabetical order, Marcus Brazil, Jens Egeblad, Stinus Lindgreen, Allan Odgaard, Doreen A. Thomas, Pawel Winter, Christian Wulff-Nilsen, and Martin Zachariasen. In particular, I am very grate- ful for the guidance of my supervisor, Martin, and for the very inspiring lunch conversations with Marcus in the Lygon Street caf´esin Melbourne — especially since this was in a time where every alley seemed to be blind. Benny Kjær Nielsen November, 2007 1Technische Universit¨atBerlin, Institut f¨urMathematik Contents Introduction 1 Nesting Problems 5 A Fast neighborhood search for two- and three-dimensional nesting problems 31 Published in the European Journal of Operational Research, 2007 B An efficient solution method for relaxed variants of the nesting problem 59 Published in Theory of Computing 2007 (Proceedings of the Thirteenth Computing: The Australasian Theory Symposium) C Translational packing of arbitrary polytopes 75 Submitted D Using directed local translations to solve the nesting problem with free orientation of shapes 107 Submitted Steiner Tree Problems 129 E On the location of Steiner points in uniformly-oriented Steiner trees 141 Published in Information Processing Letters, 2002 F Rotationally optimal spanning and Steiner trees in uniform orientation metrics 149 Published in Computational Geometry: Theory and Applications, 2004 G Deferred path heuristic for phylogenetic trees revisited 163 Published in CompBioNets 2005: Algorithmics and Computational Methods for Biochemical and Evolutionary Networks H A novel approach to phylogenetic trees: d-dimensional geometric Steiner trees 179 To appear in a special issue of Networks Introduction Introduction Two major subjects are treated in this thesis; nesting problems and Steiner tree problems. In short, nesting problems are packing problems in which the utilization of some material (or space) is to be maximized, and Steiner tree problems are tree construction problems in which the length of a tree connecting a set of points/vertices is to be minimized. These subjects are introduced separately in this thesis since they have little in common, but both nesting problems and Steiner tree problems are, in general, NP-hard. Exceptions do exist, e.g., the design of microprocessors involves packing submodules while minimizing wiring length. Each of the introductions is followed by four papers on the subject. The following is a short overview of the two subjects and the related papers. The emphasis in this thesis is on efficient algorithms and problem specific heuristics which can help solve hard optimization problems. The nesting papers are mainly concerned with variations of a translation algorithm which has proven to be a very useful geometric tool for solving nesting problems. Given some placement of the shapes (represented as polygons in 2D), the translation algorithm solves the problem of moving a shape in an axis-aligned direction to a position which minimizes its overlap with all other shapes. The first paper by Egeblad et al. [4]A introduces the translation algorithm and shows that state-of-the-art results can be obtained when it is applied to two-dimensional packing prob- lems with irregular shapes. They also generalize the translation algorithm to efficiently handle three-dimensional packing problems, but they give no proof of its correctness. Nielsen [5]B showed that the translation algorithm can be modified to handle packing problems in which the solution is going to be repeated (like a tiling), and Egeblad et al. [3]C formally proved the correctness of the translation algorithm in three dimensions while also generalizing it to an arbitrary number of dimensions. Finally, Nielsen et al. [6]D described how the translation algorithm can be implemented efficiently for translations in non-axis-aligned directions. This is then utilized for a novel approach for solving nesting problems in which free orientation of the shapes is allowed. The introduction to nesting problems contains additional material on the subject of measures of overlap. An extended version of this material is under preparation for a future nesting related paper. The introduction also discusses other geometric techniques used for solving nesting problems in the existing literature. In all of the nesting papers, the same meta-heuristic technique is applied (guided local search). This works well, but it does not mean that it is the only possible choice. It emphasizes the fact that all of the papers are concerned with algorithms and heuristics which are closely related to the geometric challenges of the problem. The subject of optimizing the parameters of a sophisticated meta-heuristic has not been addressed in any of the papers. This is partly motivated by the belief that the choice of meta-heuristic is much less important than the development of good problem specific algorithms and heuristics. The Steiner tree related papers treat more diverse problems. The first paper by Nielsen et al. [7]E is the only paper in this thesis which does not contain any computational ex- periments. The paper is concerned with the location of so-called Steiner points in uniform orientation metrics. In short, the problem is to construct a minimum length tree connecting a set of points in the plane with the restriction that all edges adhere to a finite set of uniformly distributed orientations. The results in this and other papers were later used to implement an exact algorithm for solving very large problem instances, see Nielsen et al. [8]. The length of such minimum length Steiner trees change when the given set of points is rotated. Finding 1 References the optimal orientation of a given set of points is the subject of the paper by Brazil et al. [2]F. The motivation of studying Steiner trees in uniform orientation metrics is an application in VLSI design, i.e., the design of integrated circuits for which the routing wires are restricted to some small set of orientations. Another application of Steiner trees can be found in the area of computational biology. Given a set of species, the problem is to construct a tree describing their evolutionary history. This is most often described as a phylogenetic tree. It is not necessarily a Steiner tree problem, because the exact problem definition depends on the model of evolution assumed. Nevertheless, it can be stated as a Steiner tree problem. Based on DNA sequences from the species, Nielsen et al. [9]G assume that the problem can be modeled as the problem of finding a Steiner tree in a very large graph. The nodes of the graph represent all possible DNA sequences up to some fixed length and the edges represent possible mutations of DNA sequences. The problem is then to find the shortest tree connecting the nodes in the graph which represent the species. Brazil et al. [1]H approach the problem in a completely different manner, but they also end up with a Steiner tree problem. Based on some measure of pairwise distances between the species, a transformation is done to represent the species by points in a high-dimensional Euclidean space. The problem is then to find a minimum length Steiner tree connecting all of the points. Eight papers are included in the thesis; four nesting related papers and four Steiner tree related papers. Six of these papers are either published or in press and two papers have been submitted recently. The papers have been adapted to follow a uniform style for this thesis. This includes changes to the size of figures, the layout of tables and the style of the references.
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