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Book Geometric Approximation Algorithms

Download or read book Geometric Approximation Algorithms written by Sariel Har-Peled and published by American Mathematical Soc.. This book was released on 2011 with total page 378 pages. Available in PDF, EPUB and Kindle. Book excerpt: Exact algorithms for dealing with geometric objects are complicated, hard to implement in practice, and slow. Over the last 20 years a theory of geometric approximation algorithms has emerged. These algorithms tend to be simple, fast, and more robust than their exact counterparts. This book is the first to cover geometric approximation algorithms in detail. In addition, more traditional computational geometry techniques that are widely used in developing such algorithms, like sampling, linear programming, etc., are also surveyed. Other topics covered include approximate nearest-neighbor search, shape approximation, coresets, dimension reduction, and embeddings. The topics covered are relatively independent and are supplemented by exercises. Close to 200 color figures are included in the text to illustrate proofs and ideas.

Book Handbook of Combinatorial Optimization

Download or read book Handbook of Combinatorial Optimization written by Ding-Zhu Du and published by Springer Science & Business Media. This book was released on 2006-08-18 with total page 395 pages. Available in PDF, EPUB and Kindle. Book excerpt: This is a supplementary volume to the major three-volume Handbook of Combinatorial Optimization set. It can also be regarded as a stand-alone volume presenting chapters dealing with various aspects of the subject in a self-contained way.

Book Approximation Algorithms for Combinatorial Optimization

Download or read book Approximation Algorithms for Combinatorial Optimization written by Klaus Jansen and published by Springer. This book was released on 2003-06-30 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2002, held in Rome, Italy in September 2002. The 20 revised full papers presented were carefully reviewed and selected from 54 submissions. Among the topics addressed are design and analysis of approximation algorithms, inapproximability results, online problems, randomization techniques, average-case analysis, approximation classes, scheduling problems, routing and flow problems, coloring and partitioning, cuts and connectivity, packing and covering, geometric problems, network design, and applications to game theory and other fields.

Book Approximation Algorithms for Combinatorial Optimization

Download or read book Approximation Algorithms for Combinatorial Optimization written by Klaus Jansen and published by Springer. This book was released on 2003-07-31 with total page 290 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the Third International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2000, held in Saarbrcken, Germany in September 2000. The 22 revised full papers presented together with four invited contributions were carefully reviewed and selected from 68 submissions. The topics dealt with include design and analysis of approximation algorithms, inapproximibility results, on-line problems, randomization techniques, average-case analysis, approximation classes, scheduling problems, routing and flow problems, coloring and partitioning, cuts and connectivity, packing and covering, geometric problems, network design, and various applications.

Book Iterative Methods in Combinatorial Optimization

Download or read book Iterative Methods in Combinatorial Optimization written by Lap Chi Lau and published by Cambridge University Press. This book was released on 2011-04-18 with total page 255 pages. Available in PDF, EPUB and Kindle. Book excerpt: With the advent of approximation algorithms for NP-hard combinatorial optimization problems, several techniques from exact optimization such as the primal-dual method have proven their staying power and versatility. This book describes a simple and powerful method that is iterative in essence and similarly useful in a variety of settings for exact and approximate optimization. The authors highlight the commonality and uses of this method to prove a variety of classical polyhedral results on matchings, trees, matroids and flows. The presentation style is elementary enough to be accessible to anyone with exposure to basic linear algebra and graph theory, making the book suitable for introductory courses in combinatorial optimization at the upper undergraduate and beginning graduate levels. Discussions of advanced applications illustrate their potential for future application in research in approximation algorithms.

Book The Design of Approximation Algorithms

Download or read book The Design of Approximation Algorithms written by David P. Williamson and published by Cambridge University Press. This book was released on 2011-04-26 with total page 517 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discrete optimization problems are everywhere, from traditional operations research planning (scheduling, facility location and network design); to computer science databases; to advertising issues in viral marketing. Yet most such problems are NP-hard; unless P = NP, there are no efficient algorithms to find optimal solutions. This book shows how to design approximation algorithms: efficient algorithms that find provably near-optimal solutions. The book is organized around central algorithmic techniques for designing approximation algorithms, including greedy and local search algorithms, dynamic programming, linear and semidefinite programming, and randomization. Each chapter in the first section is devoted to a single algorithmic technique applied to several different problems, with more sophisticated treatment in the second section. The book also covers methods for proving that optimization problems are hard to approximate. Designed as a textbook for graduate-level algorithm courses, it will also serve as a reference for researchers interested in the heuristic solution of discrete optimization problems.

Book Approximation Algorithms for Combinatorial Optimization

Download or read book Approximation Algorithms for Combinatorial Optimization written by Klaus Jansen and published by Springer Science & Business Media. This book was released on 2002-09-02 with total page 280 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the refereed proceedings of the 5th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2002, held in Rome, Italy in September 2002. The 20 revised full papers presented were carefully reviewed and selected from 54 submissions. Among the topics addressed are design and analysis of approximation algorithms, inapproximability results, online problems, randomization techniques, average-case analysis, approximation classes, scheduling problems, routing and flow problems, coloring and partitioning, cuts and connectivity, packing and covering, geometric problems, network design, and applications to game theory and other fields.

Book Approximation  Randomization and Combinatorial Optimization  Algorithms and Techniques

Download or read book Approximation Randomization and Combinatorial Optimization Algorithms and Techniques written by Chandra Chekuri and published by Springer Science & Business Media. This book was released on 2005-08-08 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the joint refereed proceedings of the 8th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2005 and the 9th International Workshop on Randomization and Computation, RANDOM 2005, held in Berkeley, CA, USA in August 2005. The volume contains 41 carefully reviewed papers, selected by the two program committees from a total of 101 submissions. Among the issues addressed are design and analysis of approximation algorithms, hardness of approximation, small space and data streaming algorithms, sub-linear time algorithms, embeddings and metric space methods, mathematical programming methods, coloring and partitioning, cuts and connectivity, geometric problems, game theory and applications, network design and routing, packing and covering, scheduling, design and analysis of randomized algorithms, randomized complexity theory, pseudorandomness and derandomization, random combinatorial structures, random walks/Markov chains, expander graphs and randomness extractors, probabilistic proof systems, random projections and embeddings, error-correcting codes, average-case analysis, property testing, computational learning theory, and other applications of approximation and randomness.

Book Approximation  Randomization  and Combinatorial Optimization  Algorithms and Techniques

Download or read book Approximation Randomization and Combinatorial Optimization Algorithms and Techniques written by Moses Charikar and published by Springer Science & Business Media. This book was released on 2007-08-07 with total page 636 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the joint refereed proceedings of the 10th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2007 and the 11th International Workshop on Randomization and Computation, RANDOM 2007, held in Princeton, NJ, USA, in August 2007. The 44 revised full papers presented were carefully reviewed and selected from 99 submissions. Topics of interest covered by the papers are design and analysis of approximation algorithms, hardness of approximation, small space and data streaming algorithms, sub-linear time algorithms, embeddings and metric space methods, mathematical programming methods, coloring and partitioning, cuts and connectivity, geometric problems, game theory and applications, network design and routing, packing and covering, scheduling, design and analysis of randomized algorithms, randomized complexity theory, pseudorandomness and derandomization, random combinatorial structures, random walks/Markov chains, expander graphs and randomness extractors, probabilistic proof systems, random projections and embeddings, error-correcting codes, average-case analysis, property testing, computational learning theory, and other applications of approximation and randomness.

Book Approximation  Randomization and Combinatorial Optimization  Algorithms and Techniques

Download or read book Approximation Randomization and Combinatorial Optimization Algorithms and Techniques written by Ashish Goel and published by Springer. This book was released on 2008-08-28 with total page 614 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume contains the papers presented at the 11th International Wo- shop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2008) and the 12th International Workshop on Randomization and Computation (RANDOM 2008), which took place concurrently at the MIT (M- sachusetts Institute of Technology) in Boston, USA, during August 25–27, 2008. APPROX focuses on algorithmic and complexity issues surrounding the development of e?cient approximate solutions to computationally di?cult problems, and was the 11th in the series after Aalborg (1998), Berkeley (1999), Saarbru ̈cken (2000), Berkeley (2001), Rome (2002), Princeton (2003), Cambridge (2004), Berkeley (2005), Barcelona (2006), and Princeton (2007). RANDOM is concerned with applications of randomness to computational and combinatorial problems, and was the 12th workshop in the series following Bologna (1997), Barcelona (1998), Berkeley (1999), Geneva (2000), Berkeley (2001), Harvard (2002), Princeton (2003), Cambridge (2004), Berkeley (2005), Barcelona (2006), and Princeton (2007). Topics of interest for APPROX and RANDOM are: design and analysis of - proximation algorithms, hardness of approximation, small space, sub-linear time, streaming, algorithms, embeddings and metric space methods, mathematical programming methods, combinatorial problems in graphs and networks, game t- ory, markets, economic applications, geometric problems, packing, covering, scheduling, approximate learning, design and analysis of randomized algorithms, randomized complexity theory, pseudorandomness and derandomization, random combinatorial structures, random walks/Markov chains, expander graphs and randomness extractors, probabilistic proof systems, random projections and - beddings, error-correcting codes, average-case analysis, property testing, com- tational learning theory, and other applications of approximation and randomness.

Book Approximation Algorithms for Covering Problems

Download or read book Approximation Algorithms for Covering Problems written by Christos Koufogiannakis and published by . This book was released on 2009 with total page 80 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Book Numerical Linear Approximation in C

Download or read book Numerical Linear Approximation in C written by Nabih Abdelmalek and published by CRC Press. This book was released on 2008-05-19 with total page 964 pages. Available in PDF, EPUB and Kindle. Book excerpt: Illustrating the relevance of linear approximation in a variety of fields, Numerical Linear Approximation in C presents a unique collection of linear approximation algorithms that can be used to analyze, model, and compress discrete data. Developed by the lead author, the algorithms have been successfully applied to several engineering proje

Book Approximation  Randomization and Combinatorial Optimization  Algorithms and Techniques

Download or read book Approximation Randomization and Combinatorial Optimization Algorithms and Techniques written by Klaus Jansen and published by Springer Science & Business Media. This book was released on 2004-08-11 with total page 436 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the joint refereed proceedings of the 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2004 and the 8th International Workshop on Randomization and Computation, RANDOM 2004, held in Cambridge, MA, USA in August 2004. The 37 revised full papers presented were carefully reviewed and selected from 87 submissions. Among the issues addressed are design and analysis of approximation algorithms, inapproximability results, approximation classes, online problems, graph algorithms, cuts, geometric computations, network design and routing, packing and covering, scheduling, game theory, design and analysis of randomised algorithms, randomized complexity theory, pseudorandomness, derandomization, probabilistic proof systems, error-correcting codes, and other applications of approximation and randomness.

Book Mathematical Foundations of Computer Science 2012

Download or read book Mathematical Foundations of Computer Science 2012 written by Branislav Rovan and published by Springer. This book was released on 2012-08-01 with total page 852 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume constitutes the refereed proceedings of the 37th International Symposium on Mathematical Foundations of Computer Science, MFCS 2012, held in Bratislava, Slovakia, in August 2012. The 63 revised full papers presented together with 8 invited talks were carefully reviewed and selected from 162 submissions. Topics covered include algorithmic game theory, algorithmic learning theory, algorithms and data structures, automata, formal languages, bioinformatics, complexity, computational geometry, computer-assisted reasoning, concurrency theory, databases and knowledge-based systems, foundations of computing, logic in computer science, models of computation, semantics and verification of programs, and theoretical issues in artificial intelligence.

Book Proceedings of the Eighteenth Annual ACM SIAM Symposium on Discrete Algorithms

Download or read book Proceedings of the Eighteenth Annual ACM SIAM Symposium on Discrete Algorithms written by Hal Gabow and published by Society for Industrial & Applied. This book was released on 2007 with total page 1317 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discrete mathematics and graph theory, including combinatorics, combinatorial optimization and networks Preface Acknowledgments Region-Fault Tolerant Geometric Spanners, M. A. Abam, M. de Berg, M. Farshi, and J. Gudmundsson A PTAS for TSP with Neighborhoods among Fat Regions in the Plane, Joseph S. B. Mitchell Optimal Dynamic Vertical Ray Shooting in Rectilinear Planar Subdivisions, Yoav Giyora and Haim Kaplan Squarepants in a Tree: Sum of Subtree Clustering and Hyperbolic Pants Decomposition, David Eppstein A Near Linear Time Constant Factor Approximation for Euclidean Bichromatic Matching (Cost), Piotr Indyk Compacting Cuts: A New Linear Formulation for Minimum Cut, Robert D. Carr, Goran Konjevod, Greg Little, Venkatesh Natarajan, and Ojas Parekh Linear Programming Relaxations of Maxcut, Wenceslas Fernandez de la Vega and Claire Kenyon-Mathieu Near-Optimal Algorithms for Maximum Constraint Satisfaction Problems, Moses Charikar, Konstantin Makarychev, and Yury Makarychev Improved Bounds for the Symmetric Rendezvous Value on the Line, Qiaoming Han, Donglei Du, Juan Vera, and Luis F. Zuluaga Efficient Solutions to Relaxations of Combinatorial Problems with Submodular Penalties via the Lovász Extension and Non-smooth Convex Optimization, Fabián A. Chudak and Kiyohito Nagano Multiple Source Shortest Paths in a Genus g Graph, Sergio Cabello and Erin W. Chambers Obnoxious Centers in Graphs, Sergio Cabello and Günter Rote Maximum Matching in Graphs with an Excluded Minor, Raphael Yuster and Uri Zwick Faster Dynamic Matchings and Vertex Connectivity, Piotr Sankowski Efficient Algorithms for Computing All Low s-t Edge Connectivities and Related Problems, Ramesh Hariharan, Telikepalli Kavitha, and Debmalya Panigrahi Analytic Combinatorics—A Calculus of Discrete Structures, Philippe Flajolet Equilibria in Online Games,Roee Engelberg and Joseph (Seffi) Naor The Approximation Complexity of Win-Lose Games, Xi Chen, Shang-Hua Teng, and Paul Valiant Convergence to Approximate Nash Equilibria in Congestion Games, Steve Chien and Alistair Sinclair Efficient Contention Resolution Protocols for Selfish Agents, Amos Fiat, Yishay Mansour, and Uri Nadav Strong Price of Anarchy,Nir Andelman, Michal Feldman, and Yishay Mansour Better Online Buffer Management , Fei Li, Jay Sethuraman, and Clifford Stein Considering Suppressed Packets Improves Buffer Management in QoS Switches, Matthias Englert and Matthias Westermann Instability of FIFO in the Permanent Sessions Model at Arbitrarily Small Network Loads, Matthew Andrews On the Separation and Equivalence of Paging Strategies,Spyros Angelopoulos, Reza Dorrigiv, and Alejandro López-Ortiz Pull-Based Data Broadcast with Dependencies: Be Fair to Users, Not to Items,Julien Robert and Nicolas Schabanel Improved Bounds for the Online Steiner Tree Problem in Graphs of Bounded Edge-Asymmetry, Spyros Angelopoulos Minimizing Movement, Erik D. Demaine, MohammadTaghi Hajiaghayi, Hamid Mahini, Amin S. Sayedi-Roshkhar, Shayan Oveisgharan, and Morteza Zadimoghaddam Optimization Problems in Multiple-Interval Graphs, Ayelet Butman, Danny Hermelin, Moshe Lewenstein, and Dror Rawitz Approximation Algorithms via Contraction Decomposition, Erik D. Demaine, MohammadTaghi Hajiaghayi, and Bojan Mohar A 1.875–Approximation Algorithm for the Stable Marriage Problem, Kazuo Iwama, Shuichi Miyazaki, and Naoya Yamauchi Improved Algorithms for Path, Matching, and Packing Problems, Jianer Chen, Songjian Lu, Sing-Hoi Sze, and Fenghui Zhang Model-Driven Optimization Using Adaptive Probes, Sudipto Guha and Kamesh Munagala Estimating the Sortedness of a Data Stream, Parikshit Gopalan, T.S. Jayram, Robert Krauthgamer, and Ravi Kumar A Near-Optimal Algorithm for Computing the Entropy of a Stream, Amit Chakrabarti, Graham Cormode, and Andrew McGregor The Communication and Streaming Complexity of Computing the Longest Common and Increasing Subsequences, Xiaoming Sun and David P. Woodruff Efficient Aggregation Algorithms for Probabilistic Data, T.S. Jayram, Satyen Kale, and Erik Vee An Unbiased Pointing Operator for Unlabeled Structures, with Applications to Counting and Sampling, Manuel Bodirsky, Eric Fusy, Mihyun Kang, and Stefan Vigerske Approximating Entropy from Sublinear Samples, Mickey Brautbar and Alex Samorodnitsky Torpid Mixing of Local Markov Chains on 3-Colorings of the Discrete Torus, David Galvin and Dana Randall Probabilistic Analysis of Linear Programming Decoding, Constantinos Daskalakis, Alexandros G. Dimakis, Richard M. Karp, and Martin J. Wainwright Scrambling Adversarial Errors Using Few Random Bits, Optimal Information Reconciliation, and Better Private Codes, Adam Smith Deterministic Pivoting Algorithms for Constrained Ranking and Clustering Problems, Anke van Zuylen, Rajneesh Hegde, Kamal Jain, and David P. Williamson Aggregation of Partial Rankings, p-Ratings and Top-m Lists, Nir Ailon Algorithms and Incentives for Robust Ranking, Rajat Bhattacharjee and Ashish Goel Matroids, Secretary Problems, and Online Mechanisms, Moshe Babaioff, Nicole Immorlica, and Robert Kleinberg An Algebraic Algorithm for Weighted Linear Matroid Intersection, Nicholas J. A. Harvey An Elementary Construction of Constant-Degree Expanders, Noga Alon, Oded Schwartz, and Asaf Shapira The k-Orientability Thresholds for G_{n,p}, Daniel Fernholz and Vijaya Ramachandran The Random Graph Threshold for k-Orientiability and a Fast Algorithm for Optimal Multiple-Choice Allocation, Julie Anne Cain, Peter Sanders, and Nick Wormald On Extremal Subgraphs of Random Graphs, Graham Brightwell, Konstantinos Panagiotou, and Angelika Steger Online Vertex Colorings of Random Graphs without Monochromatic Subgraphs, Martin Marciniszyn and Reto Spöhel On Testable Properties in Bounded Degree Graphs, Artur Czumaj and Christian Sohler Embedding Metrics into Ultrametrics and Graphs into Spanning Trees with Constant Average Distortion, Ittai Abraham, Yair Bartal, and Ofer Neiman Approximation Algorithms for Embedding General Metrics into Trees, Mihai Badoiu, Piotr Indyk, and Anastasios Sidiropoulos Embedding into $l^2_{infty}$ Is Easy, Embedding into $l^3_{infty}$ Is NP-Complete, Jeff Edmonds Efficient Subspace Approximation Algorithms, Nariankadu D. Shyamalkumar and Kasturi Varadarajan A Divide and Conquer Algorithm for d-Dimensional Arrangement, Moses Charikar, Konstantin Makarychev, and Yury Makarychev Resilient Search Trees, Irene Finocchi, Fabrizio Grandoni, and Giuseppe F. Italiano Randomization Does Not Help Searching Predecessors, Mihai Patrascu and Mikkel Thorup Dynamic Weighted Ancestors, Tsvi Kopelowitz and Moshe Lewenstein Ultra-succinct Representation of Ordered Trees, Jesper Jansson, Kunihiko Sadakane, and Wing-Kin Sung Tree Exploration with Logarithmic Memory, Leszek Gasieniec, Andrzej Pelc, Tomasz Radzik, and Xiaohui Zhang Testing for a Theta, Maria Chudnovsky and Paul Seymour Deterministic Rendezvous, Treasure Hunts and Strongly Universal Exploration Sequences, Amnon Ta-Shma and Uri Zwick Planar Graphs Are in 1-STRING, J. Chalopin, D. Gonçalves, and P. Ochem On the Bandwidth Conjecture for 3-Colourable Graphs, Julia Böttcher, Mathias Schacht, and Anusch Taraz Sandpile Transience on the Grid is Polynomially Bounded, László Babai and Igor Gorodezky Digraph Measures: Kelly Decompositions, Games, and Orderings, Paul Hunter and Stephan Kreutzer Approximating the Spanning Star Forest Problem and Its Applications to Genomic Sequence Alignment, C. Thach Nguyen, Jian Shen, Minmei Hou, Li Sheng, Webb Miller, and Louxin Zhang Fast Elimination of Redundant Linear Equations and Reconstruction of Recombination-Free Mendelian Inheritance on a Pedigree, Jing Xiao, Lan Liu, Lirong Xia, and Tao Jiang Whole Genome Duplications, Multi-break Rearrangements, and Genome Halving Problem, Max A. Alekseyev and Pavel A. Pevzner Succinct Indexes for Strings, Binary Relations and Multi-labeled Trees, Jérémy Barbay, Meng He, J. Ian Munro, and S. Srinivasa Rao A Simple Storage Scheme for Strings Achieving Entropy Bounds, Paolo Ferragina and Rossano Venturini A Network Formation Game for Bipartite Exchange Economies, Eyal Even-Dar, Michael Kerans, and Siddharth Suri Cheap Labor Can Be Expensive, Ning Chen and Anna R. Karlin Buying Cheap Is Expensive: Hardness of Non-parametric Multi-product Pricing, Patrick Briest and Piotr Krysta Dynamic Pricing for Impatient Bidders, Nikhil Bansal, Ning Chen, Neva Cherniavsky, Atri Rudra, Baruch Schieber, and Maxim Sviridenko Designing and Learning Optimal Finite Support Auctions, Edith Elkind On Bregman Voronoi Diagrams, Frank Nielsen, Jean-Daniel Boissonnat, and Richard Nock Zone Diagrams: Existence, Uniqueness, and Algorithmic Challenge, Tetsuo Asano, Jirí Matoušek, and Takeshi Tokuyama Approximate Shortest Paths in Anisotropic Regions, Siu-Wing Cheng, Hyeon-Suk Na, Antoine Vigneron, and Yajun Wang On Bounded Leg Shortest Paths Problems, Liam Roditty and Michael Segal Counting Colors in Boxes, Haim Kaplan, Natan Rubin, Micha Sharir, and Elad Verbin Energy Efficient Online Deadline Scheduling, Ho-Leung Chan, Wun-Tat Chan, Tak-Wah Lam, Lap-Kei Lee, Kin-Sum Mak, and Prudence W. H. Wong Speed Scaling for Weighted Flow Time, Nikhil Bansal, Kirk Pruhs, and Cliff Stein Path-Independent Load Balancing with Unreliable Machines, James Aspnes, Yang Richard Yang, and Yitong Yin Layered Multicast Scheduling for the L_infinity Objective, Qingbo Cai and Vincenzo Liberatore Lower Bounds on Average-Case Delay for Video-on-Demand Broadcast Protocols, Wei-Lung Dustin Tseng and David Kirkpatrick Maximum s-t-Flow with k Crossings in $O(k^3 n log n)$ Time, Jan M. Hochstein and Karsten Weihe Matrix Scaling by Network Flow, Günter Rote and Martin Zachariasen Single Source Multiroute Flows and Cuts on Uniform Capacity Networks, Henning Bruhn, Jakub Cerný, Alexander Hall, and Petr Kolman Island Hopping and Path Colouring with Applications to WDM Network Design, Andrew McGregor and Bruce Shepherd Maximum Independent Sets in Graphs of Low Degree,