The three-dimensional bin packing problem is the problem of orthogonally packing a set of boxes into a minimum number of three-dimensional bins. In this paper we present a heuristic algorithm based on Guided Local Search (GLS). Starting with an upper bound on the number of bins obtained by a greedy heuristic, the presented algorithm iteratively decreases the number of bins, each time searching for a feasible packing of the boxes using GLS. The process terminates when a given time limit has been reached or the upper bound matches a precomputed lower bound. The algorithm can also be applied to two-dimensional bin packing problems by having a constant depth for all boxes and bins. Computational experiments are reported for two-and three-dimensional instances with up to 200 boxes, and the results are compared with those obtained by heuristics and exact methods from the literature.

No abstract

No abstract

The history of the Euclidean Steiner tree problem, which is the problem of constructing a shortest possible network interconnecting a set of given points in the Euclidean plane, goes back to Gergonne in the early 19th century. We present a detailed account of the mathematical contributions of some of the earliest papers on the Euclidean Steiner tree problem. Furthermore, we link these initial contributions with results from the recent literature on the problem.

Abstract: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.

The GeoSteiner software package has for more than 10 years been the fastest (publicly available) program for computing exact solutions to Steiner tree problems in the plane. The computational study by Warme, Winter and Zachariasen, published in 2000, documented the performance of the GeoSteiner approach -allowing the exact solution of Steiner tree problems with more than a thousand terminals. Since then, a number of algorithmic enhancements have improved the performance of the software package significantly. In this computational study we run the current code on the largest problem instances from the 2000-study, and on a number of larger problem instances. The computational study is performed using the commercial GeoSteiner 4.0 code base, and the performance is compared to the publicly available GeoSteiner 3.1 code base as well as the code base from the 2000-study.

Abstract:The fastest exact algorithm (in practice) for the rectilinear Steiner tree problem in the plane uses a two-phase scheme: First, a small but sufficient set of full Steiner trees (FSTs) is generated and then a Steiner minimum tree is constructed from this set by using simple backtrack search, dynamic programming, or an integer programming formulation. FST generation methods can be seen as problemreduction algorithms and are also useful as a first step in providing good upper and lower bounds for large instances. Currently, the time needed to generate FSTs poses a significant overhead for FST-based exact algorithms. In this paper, we present a very efficient algorithm for the rectilinear FST generation problem which removes this overhead completely. Based on information obtained in a preprocessing phase, the new algorithm ''grows'' FSTs while applying several new and important optimality conditions. For randomly generated instances, approximately 4n FSTs are generated (where n is the number of terminals). The observed running time is quadratic and the FSTs for a 10,000 terminal instance can, on average, be generated within 5 minutes.

No abstract