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- Jack Edmonds
- Combinatorial Optimization
- Combinatorial optimization
- Combinatorial optimization. Polyhedra and efficiency.

Graham, La Jolla B. Korte, Bonn L. Lovsz, Budapest A. Wigderson, Princeton G. Library of Congress Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress.

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, , in its current version, and permission for use must always be obtained from Springer-Verlag.

Violations are liable for prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks etc. The book by Gene Lawler from was the first of a series of books all en- titled Combinatorial Optimization, some embellished with a subtitle: Net- works and Matroids, Algorithms and Complexity, Theory and Algorithms.

Why adding another book to this illustrious series? The justification is con- tained in the subtitle of the present book, Polyhedra and Eciency. This is shorthand for Polyhedral Combinatorics and Ecient Algorithms. Pioneered by the work of Jack Edmonds, polyhedral combinatorics has proved to be a most powerful, coherent, and unifying tool throughout com- binatorial optimization.

Not only it has led to ecient that is, polynomial- time algorithms, but also, conversely, ecient algorithms often imply poly- hedral characterizations and related min-max relations. It makes the two sides closely intertwined. We aim at oering both an introduction to and an in-depth survey of poly- hedral combinatorics and ecient algorithms.

Within the span of polyhedral methods, we try to present a broad picture of polynomial-time solvable com- binatorial optimization problems more precisely, of those problems that have been proved to be polynomial-time solvable.

Next to that, we go into a few prominent NP-complete problems where polyhedral methods were suc- cessful in obtaining good bounds and approximations, like the stable set and the traveling salesman problem. By definition, being in P means being solvable by a deterministic se- quential polynomial-time algorithm, and in our discussions of algorithms and complexity we restrict ourselves mainly to this characteristic.

As a conse- quence, we do not cover but yet occasionally touch or outline the important work on approximative, randomized, and parallel algorithms and complex- ity, areas that are recently in exciting motion. We also neglect applications, modelling, and computational methods for NP-complete problems. Advanced data structures are treated only moderately. Other underexposed areas in- clude semidefinite programming and graph decomposition. This all just to keep size under control.

VI Preface. Although most problems that come up in practice are NP-complete or worse, recognizing those problems that are polynomial-time solvable can be very helpful: polynomial-time and polyhedral methods may be used in pre- processing, in obtaining approximative solutions, or as a subroutine, for in- stance to calculate bounds in a branch-and-bound method. A good under- standing of what is in the polynomial-time tool box is essential also for the NP-hard problem solver.

This book is divided into eight main parts, each discussing an area where polyhedral methods apply: I. Paths and Flows II. Nonbipartite Matching and Covering IV. Matroids and Submodular Functions V. Trees, Branchings, and Connectors VI. Each of the eight parts starts with an elementary exposition of the basic results in the area, and gradually evolves to the more elevated regions.

Sub- sections in smaller print go into more specialized topics. We also oer several references for further exploration of the area. Although we give elementary introductions to the various areas, this book might be less satisfactory as an introduction to combinatorial optimization. Some mathematical maturity is required, and the general level is that of graduate students and researchers.

Yet, parts of the book may serve for un- dergraduate teaching. The book does not oer exercises, but, to stimulate research, we collect open problems, questions, and conjectures that are mentioned throughout this book, in a separate section entitled Survey of Problems, Questions, and Conjectures in Volume C.

It is not meant as a complete list of all open problems that may live in the field, but only of those mentioned in the text. We assume elementary knowledge of and familiarity with graph theory, with polyhedra and linear and integer programming, and with algorithms and complexity.

To support the reader, we survey the knowledge assumed in the introductory chapters, where we also give additional background refer- ences. These chapters are meant mainly just for consultation, and might be less attractive to read from front to back.

Some less standard notation and terminology are given on the inside back cover of this book. Wiley, Chichester, This might seem a biased recommendation, but this book was partly written as a preliminary to the present book, and it covers anyway the authors knowledge on polyhedra and linear and integer programming.

Incidentally, the reader of this book will encounter a number of concepts and techniques that regularly crop up: total unimodularity, total dual inte- grality, duality, blocking and antiblocking polyhedra, matroids, submodular- ity, hypergraphs, uncrossing. It makes that the meaning of elementary is not unambiguous.

Especially for the basic results, several methods apply, and it is not in all cases obvious which method and level of generality should be chosen to give a proof. In some cases we therefore will give several proofs of one and the same theorem, just to open the perspective. While I have pursued great carefulness and precision in composing this book, I am quite sure that much room for corrections and additions has remained.

To inform the reader about them, I have opened a website at the address www. In preparing this book I have profited greatly from the support and help of many friends and colleagues, to whom I would like to express my gratitude. I am particularly much obliged to Sasha Karzanov in Moscow, who has helped me enormously by tracking down ancient publications in the former Lenin Library in Moscow and by giving explanations and interpretations of old and recent Russian papers.

I also thank Sashas sister Irina for translating Tolstos article for me. Sincere thanks are due as well to Truus W. Koopmans for sharing with me her memories and stories and sections from her late husbands war di- ary, and to Herb Scarf for his kind mediation in this. The assistance of my institute, CWI in Amsterdam, has been indispens- able in writing this book.

My special thanks go to Karin van Gemert of the CWI Library for her indefatigable eorts in obtaining rare publications from every corner of the world, always in sympathizing understanding for my of- ten extravagant requests. Also Thea de Hoog and Rick Ooteman were very helpful in this. In the technical realization of this book, I thankfully enjoyed the first- rate workmanship of the sta of Springer-Verlag in Heidelberg.

As it has turned out, it was only by gravely neglecting my family that I was able to complete this project. I am extremely grateful to Monique, Nella, and Juliette for their perpetual understanding and devoted support. Now comes the time for the pleasant fulfilment of all promises I made for when my book will be finished.

Volume A 1 Introduction. Part I: Paths and Flows Part V: Trees, Branchings, and Connectors Name Index. Subject Index. Greek graph and hypergraph functions. Introduction Combinatorial optimization searches for an optimum object in a finite collec- tion of objects. Typically, the collection has a concise representation like a graph , while the number of objects is huge more precisely, grows exponen- tially in the size of the representation like all matchings or all Hamiltonian circuits.

So scanning all objects one by one and selecting the best one is not an option. More ecient methods should be found. In the s, Edmonds advocated the idea to call a method ecient if its running time is bounded by a polynomial in the size of the representation.

Since then, this criterion has won broad acceptance, also because Edmonds found polynomial-time algorithms for several important combinatorial opti- mization problems like the matching problem. The class of polynomial-time solvable problems is denoted by P. Further relief in the landscape of combinatorial optimization was discov- ered around when Cook and Karp found out that several other promi- nent combinatorial optimization problems including the traveling salesman problem are the hardest in a large natural class of problems, the class NP.

The class NP includes most combinatorial optimization problems. Any prob- lem in NP can be reduced to such NP-complete problems. All NP-complete problems are equivalent in the sense that the polynomial-time solvability of one of them implies the same for all of them. Almost every combinatorial optimization problem has since been either proved to be polynomial-time solvable or NP-complete and none of the problems have been proved to be both.

This book focuses on those combinatorial optimization problems that have been proved to be solvable in polynomial time, that is, those that have been proved to belong to P.

Next to polynomial-time solvability, we focus on the related polyhedra and min-max relations. These three aspects have turned out to be closely related, as was shown also by Edmonds. Con- versely, an appropriate description of the polyhedron often implies the polynomial-time solvability of the associated optimization problem, by ap- plying linear programming techniques.

With the duality theorem of linear programming, polyhedral characterizations yield min-max relations, and vice versa. So the span of this book can be portrayed alternatively by those combi- natorial optimization problems that yield well-described polyhedra and min- max relations.

This field of discrete mathematics is called polyhedral combi- natorics. In the following sections we give some basic, illustrative examples. We will call w F the weight of F. In notation, we want to solve 1. We can formulate this problem equivalently as follows.

Hence problem 1. This amounts to maximizing the linear function wT x over a finite set of vectors. Therefore, the optimum value does not change if we maximize over the convex hull of these vectors: 1. The set 1.

For thousands of years, and even now in group theoretic geometry, the beautiful symmetries of a handful of polyhedra with a handful of facets have been at the center of refined mathematics. Since the advent of Turing's computers and operations research, beauty has been found in polyhedra regardless of symmetry, with facets as numerous as the stars. Linear-algebra theory is being nudged by great systems of linear inequalities as inputs. NP, means certifiable in polynomial time when true. The course will explore some of the polyhedra which have edged aside the dodecahedron. Topics of the various days are related but presentations will be independent with some reference to each other.

Graham, La Jolla B. Korte, BonnL. Lovsz, Budapest A. Wigderson, PrincetonG.

Printable version pdf. The paper also introduced the concept of using black-box recognition of independence as an algorithmic oracle. Also in those lectures he presented with Dick Karp the first polynomial-time algorithms for network flows, and asked for strongly polynomial algorithms i.

Graham, La Jolla B. Korte, Bonn L. Lovsz, Budapest A.

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Combinatorial optimization is a subfield of mathematical optimization that is related to operations research , algorithm theory , and computational complexity theory. It has important applications in several fields, including artificial intelligence , machine learning , auction theory , software engineering , applied mathematics and theoretical computer science. Combinatorial optimization is a topic that consists of finding an optimal object from a finite set of objects.

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It seems that you're in Germany. We have a dedicated site for Germany. This book offers an in-depth overview of polyhedral methods and efficient algorithms in combinatorial optimization. These methods form a broad, coherent and powerful kernel in combinatorial optimization, with strong links to discrete mathematics, mathematical programming and computer science.

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