The Steiner Tree Problem

TLDR: The Steiner Ratio Conjecture as a Maximin Problem and Effectiveness of Reductions, and Heuristics Using a Given RMST Algorithms, and two Related Results.

Abstract: Euclidean Steiner Problem. Introduction. Historical Background. Some Basic Notions. Some Basic Properties. Full Steiner Trees. Steiner Hulls and Decompositions. The Number of Steiner Topologies. Computational Complexity. Physical Models. References. Exact Algorithms. The Melzak Algorithm. A Linear-Time FST Algorithm. Two Ideas on the Melzak Algorithm. A Numberical Algorithm. Pruning. The GEOSTEINER Algorithm. The Negative Edge Algorithm. The Luminary Algorithm. References. The Steiner Ratio. Lower Bounds of rho. The Small n Case. The Variational Approach. The Steiner Ratio Conjecture as a Maximin Problem. Critical Structures. A Proof of the Steiner Ratio Conjecture. References. Heuristics. Minimal Spanning Trees. Improving the MST. Greedy Trees. An Annealing Algorithm. A Partitioning Algorithm. Few's Algorithms. A Graph Approximation Algorithm. k-Size Quasi-Steiner Trees. Other Heuristics. References. Special Terminal-Sets. Four Terminals. Cocircular Terminals. Co-path Terminals. Terminals on Lattice Points. Two Related Results. References. Generalizations. d-Dimensional Euclidean Spaces. Cost of Edge. Terminal Clusters and New Terminals. k-SMT. Obstacles. References. Steiner Problem in Networks. Introduction. Applications. Definitions. Trivial Special Cases. Problem Reformulations. Complexity. References. Reductions. Exclusion Tests. Inclusion Tests. Integration of Tests. Effectiveness of Reductions. References. Exact Algorithms. Spanning Tree Enumeration Algorithm. Degree-Constrained Tree Enumeration Algorithm. Topology Enumeration Algorithm. Dynamic Programming Algorithm. Branch-and-Bound Algorithm. Mathematical Programming Formulations. Linear Relaxations. Lagrangean Relaxations. Benders' Decomposition Algorithm. Set Covering Algorithm. Summary and Computational Experience. References. Heuristics. Path Heurisitics. Tree Heuristics. Vertex Heuristics. Contraction Heuristic. Dual Ascent Heuristic. Set Covering Heuristic. Summary and Computational Experience. References. Polynomially Sovable Cases. Series-Parallel Networks. Halin Networks. k-Planar Networks. Strongly Chordal Graphs. References. Generalizations. Steiner Trees in Directed Networks. Weighted Steiner Tree Problem. Steiner Forest Problem. Hierarchical Steiner Tree Problem. Degree-Dependent Steiner Tree Problem. Group Steiner Tree Problem. Multiple Steiner Trees Problem. Multiconnected Steiner Network Problem. Steiner Problem in Probabilistic Networks. Realization of Distance Matrices. Other Steiner-Like Problems. References. Rectilinear Steiner Problem. Introduction. Definitions. Basic Properties. A Characterization of RSMTs. Problem Reductions. Extremal Results. Computational Complexity. Exact Algorithms. References. Heuristic Algorithms. Heuristics Using a Given RMST. Heuristics Based on MST Algorithms. Computational Geometry Paradigms. Other Heuristics. References. Polynomially Solvable Cases. Terminals on a Rectangular Boundary. Rectilinearly Convex Boundary.