https://sci2s.ugr.es/sites/default/files/files/Teaching/GraduatesCourses/Metaheuristicas/Bibliography/VNS.pdf

Variable neighborhood search

Triangle is a robust implementation of two-dimensional constrained Delaunay triangulation and Ruppert's Delaunay refinement algorithm for quality mesh generation. Several implementation issues are discussed, including the choice of triangulation algorithms and data structures, the effect of several variants of the Delaunay refinement algorithm on mesh quality, and the use of adaptive exact arithmetic to ensure robustness with minimal sacrifice of speed. The problem of triangulating a planar straight line graph (PSLG) without introducing new small angles is shown to be impossible for some PSLGs, contradicting the claim that a variant of the Delaunay refinement algorithm solves this problem.

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Triangle: Engineering a 2D Quality Mesh Generator and Delaunay Triangulator

We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c > 1 and given any n nodes in ℛ2, a randomized version of the scheme finds a (1 + 1/c)-approximation to the optimum traveling salesman tour in O(n(log n)O(c)) time. When the nodes are in ℛd, the running time increases to O(n(log n)(O(√c))d-1). For every fixed c, d the running time is n · poly(logn), that is nearly linear in n. The algorithmm can be derandomized, but this increases the running time by a factor O(nd). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2-aproximation in polynomial time.We also give similar approximation schemes for some other NP-hard Euclidean problems: Minimum Steiner Tree, k-TSP, and k-MST. (The running times of the algorithm for k-TSP and k-MST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constant-factor approximation. We also give efficient approximation schemes for Euclidean Min-Cost Matching, a problem that can be solved exactly in polynomial time.All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as lp for p ≥ 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.

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Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems

The upcoming gigabit-per-second high-speed networks are expected to support a wide range of communication-intensive real-time multimedia applications. The requirement for timely delivery of digitized audio-visual information raises new challenges for next-generation integrated services broadband networks. One of the key issues is QoS routing. It selects network routes with sufficient resources for the requested QoS parameters. The goal of routing solutions is twofold: (1) satisfying the QoS requirements for every admitted connection, and (2) achieving global efficiency in resource utilization. Many unicast/multicast QoS routing algorithms have been published, and they work with a variety of QoS requirements and resource constraints. Overall, they can be partitioned into three broad classes: (1) source routing, (2) distributed routing, and (3) hierarchical routing algorithms. We give an overview of the QoS routing problem as well as the existing solutions. We present the strengths and weaknesses of different routing strategies, and outline the challenges. We also discuss the basic algorithms in each class, classify and compare them, and point out possible future directions in the QoS routing area.

An overview of quality of service routing for next-generation high-speed networks: problems and solutions

A Unified Approach for Domination Problems on Different Network Topologies Advanced Techniques for Dynamic Programming Advances in Group Testing Advances in Scheduling Problems Algebrization and Randomization Methods Algorithmic Aspects of Domination in Graphs Algorithms and Metaheuristics for Combinatorial Matrices Algorithms for the Satisfiability Problem Bin Packing Approximation Algorithms: Survey and Classification Binary Unconstrained Quadratic Optimization Problem Combinatorial Optimization Algorithms for Probe Design and Selection Problems Combinatorial Optimization in Data Mining Combinatorial Optimization Techniques for Network-based Data Mining Combinatorial Optimization Techniques in Transportation and Logistic Networks Complexity Issues on PTAS Computing Distances between Evolutionary Trees Connected Dominating Set in Wireless Networks Connections between Continuous and Discrete Extremum Problems, Generalized Systems and Variational Inequalities Coverage Problems in Sensor Networks Data Correcting Approach for Routing and Location in Networks Dual Integrality in Combinatorial Optimization Dynamical System Approaches to Combinatorial Optimization Efficient Algorithms for Geometric Shortest Path Query Problems Energy Efficiency in Wireless Networks Equitable Coloring of Graphs Faster and Space Efficient Exact Exponential Algorithms: Combinatorial and Algebraic Approaches Fault-Tolerant Facility Allocation Fractional Combinatorial Optimization Fuzzy Combinatorial Optimization Problems Geometric Optimization in Wireless Networks Gradient-Constrained Minimum Interconnection Networks Graph Searching and Related Problems Graph Theoretic Clique Relaxations and Applications Greedy Approximation Algorithms Hardness and Approximation of Network Vulnerability Job Shop Scheduling with Petri Nets Key Tree Optimization Linear Programming Analysis of Switching Networks Map of Geometric Minimal Cuts with Applications Max-Coloring Maximum Flow Problems and an NP-complete variant on Edge Labeled Graphs Modern Network Interdiction Problems and Algorithms Network Optimization Neural Network Models in Combinatorial Optimization On Coloring Problems Online and Semi-online Scheduling Online Frequency Allocation and Mechanism Design for Cognitive Radio Wireless Networks Optimal Partitions Optimization in Multi-Channel Wireless Networks Optimization Problems in Data Broadcasting Optimization Problems in Online Social Networks Optimizing Data Collection Capacity in Wireless Networks Packing Circles in Circles and Applications Partition in High Dimensional Spaces Probabilistic Verification and Non-approximability Protein Docking Problem as Combinatorial Optimization Using Beta-complex Quadratic Assignment Problems Reactive Business Intelligence: Combining the Power of Optimization with Machine Learning Reformulation-Linearization Techniques for Discrete Optimization Problems Resource Allocation Problems Rollout Algorithms for Discrete Optimization: A Survey Simplicial Methods for Approximating Fixed Point with Applications in Combinatorial Optimizations Small World Networks in Computational Neuroscience Social Structure Detection Steiner Minimal Trees: An Introduction, Parallel Computation and Future Work Steiner Minimum Trees in E^3 Tabu Search Variations of Dominating Set Problem

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Handbook of Combinatorial Optimization

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.

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The Steiner Tree Problem

We give a survey up to 1989 on the Steiner tree problems which include the four important cases of euclidean, rectilinear, graphic, phylogenetic and some of their generalizations. We also provide a rather comprehensive and up-to-date bibliography which covers more than three hundred items.

Steiner tree problems

Mesh generation and optimal triangulation, M. Bern and D. Eppstein machine proofs of geometry theorems, S.C. Chou and M. Rethi randomized geometric algorithms, K. Clarkson Voronoi diagrams and Delanney triangulations, S. Fortune the state of art on Steiner ratio problems, D-Z. Du and F. Hwang on the development of quantitative geometry from Pythagoras to Grassmann, W-Y. Hsiang computational geometry and topological network designs, J. Smith and P. Winter polar forms and triangular B-spline surfaces, H-P. Seidel algebraic foundations of computational geometry, Chee Yap.

Computing in Euclidean Geometry

LetP be a set ofn points on the euclidean plane. LetL
 s(P) andL
 m
 (P) denote the lengths of the Steiner minimum tree and the minimum spanning tree onP, respectively. In 1968, Gilbert and Pollak conjectured that for anyP,L
 s
 (P)≥(√3/2)L
 m
 (P). We provide a proof for their conjecture in this paper.

A proof of the Gilbert-Pollak conjecture on the Steiner ratio

Let P be a set of n points on the euclidean plane. Let L~(P) and L,,(P) denote the lengths of the Steiner minimum tree and the minimum spanning tree on P, respectively. In 1968, Gilbert and Pollak conjectured that for any P, Ls(P) > (Xf3/2)Lm(P). We provide a proof for their conjecture in this paper.

Frank K. Hwang

Papers

The Steiner Tree Problem

Steiner tree problems

Computing in Euclidean Geometry

A proof of the Gilbert-Pollak conjecture on the Steiner ratio

A Proof of the Gilbert-Pollak Conjecture on the