Pawel Winter

Steiner problem in networks: a survey

The problem of determining a minimum cost connected network (i.e., weighted graph) G that spans a given subset of vertices is known in the literature as the Steiner problem in networks. We survey exact algorithms and heuristics which appeared in the published literature. We also discuss problems related to the Steiner problem in networks.

Exact Algorithms for Plane Steiner Tree Problems: A Computational Study

We present a computational study of exact algorithms for the Euclidean and rectilinear Steiner tree problems in the plane. These algorithms — which are based on the generation and concatenation of full Steiner trees — are much more efficient than other approaches and allow exact solutions of problem instances with more than 2000 terminals. The full Steiner tree generation algorithms for the two problem variants share many algorithmic ideas and the concatenation part is identical (integer programming formulation solved by branch-and-cut). Performance statistics for randomly generated instances, public library instances and “difficult” instances with special structure are presented. Also, results on the comparative performance on the two problem variants are given.

Path-distance heuristics for the Steiner problem in undirected networks

An integrative overview of the algorithmic characteristics of three well-known polynomialtime heuristics for the undirected Steiner minimum tree problem:shortest path heuristic (SPH),distance network heuristic (DNH), andaverage distance heuristic (ADH) is given. The performance of thesesingle-pass heuristics (and some variants) is compared and contrasted with several heuristics based onrepetitive applications of the SPH. It is shown that two of these repetitive SPH variants generate solutions that in general are better than solutions obtained by any single-pass heuristic. The worst-case time complexity of the two new variants isO(pn3) andO(p3n2), while the worst-case time complexity of the SPH, DNH, and ADH is respectivelyO(pn2),O(m + n logn), andO(n3) wherep is the number of vertices to be spanned,n is the total number of vertices, andm is the total number of edges. However, use of few simple tests is shown to provide large reductions of problem instances (both in terms of vertices and in term of edges). As a consequence, a substantial speed-up is obtained so that the repetitive variants are also competitive with respect to running times.

An algorithm for the steiner problem in the euclidean plane

An algorithm for the exact solution of the Steiner problem in the Euclidean plane is presented. Compared with earlier algorithms, it generates full topologies in a different manner whereby the number of computations is substantially reduced. Furthermore, a large number of full Steiner trees which do not belong to the Steiner minimal tree is identified and discarded by new and efficient tests. The algorithm appears to be considerably faster than any other existing algorithm.

Euclidean Steiner minimum trees: An improved exact algorithm

The Euclidean Steiner tree problem asks for a shortest network interconnecting a set of terminals in the plane. Over the last decade, the maximum problem size solvable within 1 h (for randomly generated problem instances) has increased from 10 to approximately 50 terminals. We present a new exact algorithm, called geosteiner96. It has several algorithmic modifications which improve both the generation and the concatenation of full Steiner trees. On average, geosteiner96 solves randomly generated problem instances with 50 terminals in less than 2 min and problem instances with 100 terminals in less than 8 min. In addition to computational results for randomly generated problem instances, we present computational results for (perturbed) regular lattice instances and public library instances.

Generalized Steiner problem in series-parallel networks

Abstract The generalized Steiner problem (GSP) is concerned with the determination of a minimum cost subnetwork of a given network where some (not necessarily all) vertices satisfy certain pairwise (vertex or edge) connectivity requirements. The GSP has applications to the design of water and electricity supply networks, communication networks and other large-scale systems where connectivity requirements ensure the communication between the selected vertices when some vertices and/or edges can become inoperational due to scheduled maintenance, error, or overload. The GSP is known to be NP-complete. In this paper we show that if the subnetwork is required to be respectively biconnected and edge-biconnected, and the underlying network is series-parallel, both problems can be solved in linear time.

An Exact Algorithm for the Uniformly-Oriented Steiner Tree Problem

An exact algorithm to solve the Steiner tree problem for uniform orientation metrics in the plane is presented. The algorithm is based on the two-phase model, consisting of full Steiner tree (FST) generation and concatenation, which has proven to be very successful for the rectilinear and Euclidean Steiner tree problems. By applying a powerful canonical form for the FSTs, the set of optimal solutions is reduced considerably. Computational results both for randomly generated problem instances and VLSI design instances are provided. The new algorithm solves most problem instances with 100 terminals in seconds, and problem instances with up to 10000 terminals have been solved to optimality.

Obstacle-Avoiding Euclidean Steiner Trees in the Plane: An Exact Algorithm

The first exact algorithm for the obstacle-avoiding Euclidean Steiner tree problem in the plane (in its most general form) is presented. The algorithm uses a two-phase framework -- based on the generation and concatenation of full Steiner trees -- previously shown to be very successful for the obstacle-free case. Computational results for moderate size problem instances are given; instances with up to 150 terminals have been solved to optimality within a few hours of CPU-time.

Steiner problem in Halin networks

Abstract The Steiner problem in networks is concerned with the determination of a minimum cost subnetwork connecting some (not necessarily all) vertices of an underlying network. The decision version of the Steiner problem is known to be NP-complete. However, if the underlying network is outerplanar or series-parallel, linear time algorithms have been developed. In this paper a linear time algorithm for the Steiner problem in Halin networks is presented. This result provides another example where the recursive structure of the underlying network leads to an efficient algorithm. Furthermore, the result is of interest from the network design point of view, since Halin networks are nontrivial generalizations of tree and ring networks.

Concatenation-based greedy heuristics for the Euclidean Steiner tree problem

Abstract. We present a class of O(n log n) heuristics for the Steiner tree problem in the Euclidean plane. These heuristics identify a small number of subsets with few, geometrically close, terminals using minimum spanning trees and other well-known structures from computational geometry: Delaunay triangulations, Gabriel graphs, relative neighborhood graphs, and higher-order Voronoi diagrams. Full Steiner trees of all these subsets are sorted according to some appropriately chosen measure of quality. A tree spanning all terminals is constructed using greedy concatenation. New heuristics are compared with each other and with heuristics from the literature by performing extensive computational experiments on both randomly generated and library problem instances.