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19
Preprocessing Steiner Problems from VLSI Layout
 Networks
, 1999
"... : 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 i ..."
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Cited by 11 (4 self)
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: 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 branchandcut to solve 28 out of 32 open instances in the SteinLib, some with more than 20,000 edges. Keywords: Steiner Problem, Preprocessing, 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 bidimensional com buracos e s~ao consideradas de dif'icil resoluc~ao pelos m'etodos conhecidos. Em particular, as t'ecnicas de pr'eprocessamento desenvolvidas pra problemas de Steiner em grafos gerais n~ao funcionam bem. Este trabalho apresenta um novo procedimento de pr'eprocessamento, adaptando testes de reduc~ao j'a conhecidos na literatura tornandoos 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 branchandcut resolvesse 28 das 32 instancias de VLSI da SteinLib at'e ent~ao em em aberto. Palavraschave: Problema de Steiner, Pr'eprocessamento, Projeto de VLSI. 1
Rapid mathematical programming
, 2004
"... 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 enoug ..."
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Cited by 10 (2 self)
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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 realworld 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?
On the Implementation of MSTbased Heuristics for the Steiner Problem in Graphs
 In Proceedings of the 4th International Workshop on Algorithm Engineering and Experiments
, 2002
"... 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 experime ..."
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Cited by 7 (2 self)
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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 worstcase 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.
New Benchmark Instances for the Steiner Problem in Graphs
 IN EXTENDED ABSTRACTS OF THE 4TH METAHEURISTICS INTERNATIONAL CONFERENCE
, 2001
"... 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 heur ..."
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Cited by 6 (1 self)
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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. Stateoftheart 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.
A PartitionBased Relaxation For Steiner Trees
, 2009
"... The Steiner tree problem is a classical NPhard 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 nonnegative costs ce for all edges e ∈ E. Any tree that contains all terminals ..."
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Cited by 4 (2 self)
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The Steiner tree problem is a classical NPhard 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 nonnegative costs ce for all edges e ∈ E. Any tree that contains all terminals is called a Steiner tree; the goal is to find a minimumcost 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 LPbased approaches by showing that Robins and Zelikovsky’s algorithm can be viewed as an iterated primaldual algorithm with respect to a novel LP relaxation. The LP used in the first iteration is stronger than the wellknown bidirected cut relaxation. An instance is bquasibipartite 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
DryadOpt: BranchandBound on Distributed DataParallel Execution Engines
"... Abstract—We introduce DryadOpt, a library that enables massively parallel and distributed execution of optimization algorithms for solving hard problems. DryadOpt performs an exhaustive search of the solution space using branchandbound, by recursively splitting the original problem into many simpl ..."
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Cited by 4 (2 self)
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Abstract—We introduce DryadOpt, a library that enables massively parallel and distributed execution of optimization algorithms for solving hard problems. DryadOpt performs an exhaustive search of the solution space using branchandbound, by recursively splitting the original problem into many simpler subproblems. It uses both parallelism (at the core level) and distributed execution (at the machine level). DryadOpt provides a simple yet powerful interface to its users, who only need to implement sequential code to process individual subproblems (either by solving them in full or generating new subproblems). The parallelism and distribution are handled automatically by DryadOpt, and are invisible to the user. The distinctive feature of our system is that it is implemented on top of DryadLINQ, a distributed dataparallel execution engine similar to Hadoop and MapReduce. Despite the fact that these engines offer a constrained application model, with restricted communication patterns, our experiments show that careful design choices allow DryadOpt to scale linearly with the number of machines, with very little overhead. Keywordscombinatorial optimization; branchandbound; distributed computation; Dryad; distributed dataparallel execution engines I.
Dealing with large hidden constants: Engineering a planar steiner tree PTAS
 IN ALENEX, SIAM
, 2009
"... We present the first attempt on implementing a highly theoretical polynomialtime approximation scheme (PTAS) with huge hidden constants, namely, the PTAS for Steiner tree in planar graphs by Borradaile, Klein, and Mathieu (SODA 2007, WADS 2007). Whereas this result, and several other PTAS results o ..."
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Cited by 3 (1 self)
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We present the first attempt on implementing a highly theoretical polynomialtime approximation scheme (PTAS) with huge hidden constants, namely, the PTAS for Steiner tree in planar graphs by Borradaile, Klein, and Mathieu (SODA 2007, WADS 2007). Whereas this result, and several other PTAS results of the recent years, are of high theoretical importance, no practical applications or even implementation attempts have been known to date, due to the extremely large constants that are involved in them. We describe techniques on how to circumvent the challenges in implementing such a scheme. Our main contribution is the engineering of several details of the original algorithm to make it work in practice. With today’s limitations on processing power and space, we still have to sacrifice approximation guarantees for improved running times by choosing some parameters empirically. But our experiments show that with our choice of parameters, we do get the desired approximation ratios, suggesting that a much tighter analysis might be possible. Hence, we show that it is possible to actually implement and run this algorithm, even on large instances, already today – but under some compromises. Further improvements, both in theory and practice, might make these great theoretical works finally bear practical fruits in the future. First computational experiments with benchmark instances from SteinLib and large artificial instances well exceeded our own expectations. We demonstrate that we are able to handle instances with up to a million nodes and several hundreds of terminals in 1.5 hours on a standard PC. On the rectilinear preprocessed instances from SteinLib, we observe a monotonous improvement for smaller values of ε, with an average gap below 1 % for ε = 0.1. We compare our implementation against the wellknown batched 1Steiner heuristic and observe that on very large instances, we are able to produce comparable solutions much faster.
Orientationbased Models for {0,1,2}Survivable Network Design: Theory and Practice
 MATHEMATICAL PROGRAMMING, SERIES B
"... We consider {0,1,2}Survivable Network Design problems with nodeconnectivity constraints. In the most prominent variant, we are given an edgeweighted graph and two customer sets R1 and R2; we ask for a minimum cost subgraph that connects all customers, and guarantees twonodeconnectivity for the ..."
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Cited by 3 (1 self)
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We consider {0,1,2}Survivable Network Design problems with nodeconnectivity constraints. In the most prominent variant, we are given an edgeweighted graph and two customer sets R1 and R2; we ask for a minimum cost subgraph that connects all customers, and guarantees twonodeconnectivity for the R2 customers. We also consider an alternative of this problem, in which 2nodeconnectivity is only required w.r.t. a certain root node, and its prizecollecting variant. The central result of this paper is a novel graphtheoretical characterization of 2nodeconnected graphs via orientation properties. This allows us to derive two classes of ILP formulations based on directed graphs, one using multicommodity flow and one using cutinequalities. We prove the theoretical advantages of these directed models compared to the previously known ILP approaches. We show that our two concepts are equivalent from the polyhedral point of view. On the other hand, our experimental study shows that the cut formulation is much more powerful in practice. Moreover, we propose a collection of benchmark instances that can be used for further
Computing steiner minimum trees in Hamming metric
 In Proceedings of the seventeenth annual ACMSIAM symposium on Discrete algorithm
, 2006
"... 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 ..."
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Cited by 2 (0 self)
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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
Fast Local Search for Steiner Trees in Graphs
"... We present efficient algorithms that implement four local searches for the Steiner problem in graphs: vertex insertion, vertex elimination, keypath exchange, and keyvertex elimination. In each case, we show how to find an improving solution (or prove that none exists in the neighborhood) in O(mlog ..."
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Cited by 2 (1 self)
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We present efficient algorithms that implement four local searches for the Steiner problem in graphs: vertex insertion, vertex elimination, keypath exchange, and keyvertex 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