When preparing the first edition of this book, more than ten years ago, we tried to accomplish two objectives: it should be useful as an advanced graduate textbook, but also as a reference work for research. With each new edition we have to decide how the book can be improved further. Of course, it is less and less possible to describe the growing area comprehensively.If we included everything that we like, the book would grow beyond a single volume. Since the book is used for many courses, now even sometimes at undergraduate level, we thought that adding some classical material might be more useful than including a selection of the latest results.In this edition, we added a proof of Cayley's formula, more details on blocking flows, the new faster b-matching separation algorithm, an approximation scheme for multidimensional knapsack, and results concerning the multicommodity max-flow min-cut ratio and the sparsest cut problem. There are further small improvements in numerous places and more than 60 new exercises. Of course, we also updated the references to point to the most recent results and corrected some minor errors that were discovered.We would like to thank Takao Asano, Maxim Babenko, Ulrich Brenner, Benjamin Bolten, Christoph Buchheim, Jean Fonlupt, András Frank, Michael Gester, Stephan Held, Stefan Hougardy, Hiroshi Iida, Klaus Jansen, Alexander Karzanov, Levin Keller, Alexander Kleff, Niko Klewinghaus, Stefan Knauf, Barbara Langfeld, Jens Maßberg, Marc Pfetsch, Klaus Radke, Rabe von Randow, Tomás Salles, Jan Schneider, Christian Schulte, András Sebő, Martin Skutella, Jácint Szabó, and Simon Wedeking for valuable feedback on the previous edition.We are pleased that this book has been received so well, and further translations are on their way. Editions in Japanese, French, Italian, German, Russian, and Chinese have appeared since 2009 or are scheduled to appear soon. We hope that our book will continue to serve its purpose in teaching and research in combinatorial optimization. Bonn, September 2011Bernhard Korte and Jens Vygen Preface to the Fourth EditionWith four English editions, and translations into four other languages forthcoming, we are very happy with the development of our book. Again, we have revised, updated, and significantly extended it for this fourth edition. We have added some classical material that may have been missed so far, in particular on linear programming, the network simplex algorithm, and the max-cut problem. We have also added a number of new exercises and up-to-date references. We hope that these changes serve to make our book an even better basis for teaching and research. We gratefully acknowledge the continuous support of the Union of the German Academies of Sciences and Humanities and the NRW Academy of Sciences via the long-term research project "Discrete Mathematics and Its Applications". We also thank those who gave us feedback on the third edition, in particular Takao Asano, Christoph Bartoschek, Bert Besser, Ulrich Brenner, Jean Fonlupt, Satoru Fujishige, Marek Kar...
Preface to the First Edition sign and the Euclidean traveling salesman problem. All results are accompanied by detailed proofs.Of course, no book on combinatorial optimization can be absolutely comprehensive. Examples of topics which we mention only briefly or do not cover at all are tree-decompositions, separators, submodular flows, path-matchings, deltamatroids, the matroid parity problem, location and scheduling problems, nonlinear problems, semidefinite programming, average-case analysis of algorithms, advanced data structures, parallel and randomized algorithms, and the theory of probabilistic ally checkable proofs (we cite the PCP Theorem without proof).At the end of each chapter there are a number of exercises containing additional results and applications of the material in that chapter. Some exercises which might be more difficult are marked with an asterisk. Each chapter ends with a list of references, including texts recommended for further reading.This book arose from several courses on combinatorial optimization and from special classes on topics like polyhedral combinatorics or approximation algorithms. Thus, material for basic and advanced courses can be selected from this book.We have benefited from discussions and suggestions of many colleagues and friends andof course -from other texts on this subject. Especially we owe sincere thanks to Andras Frank, Laszlo Lovasz, Andras Recski, Alexander Schrijver and Zoltan Szigeti. Our colleagues and students in Bonn,
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