Graduate Texts in Mathematics bridge the gap between passive study and creative understanding, offering graduate-level introductions to advanced topics in mathematics. The volumes are carefully written as teaching aids and highlight characteristic features of the theory. Although these books are frequently used as textbooks in graduate courses, they are also suitable for individual study. This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. Exempted from this legal reservation are brief excerpts in connection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Duplication of this publication or parts thereof is permitted only under the provisions of the Copyright Law of the Publisher's location, in its current version, and permission for use must always be obtained from Springer. Permissions for use may be obtained through RightsLink at the Copyright Clearance Center. Violations are liable to prosecution under the respective Copyright Law. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com) PrefaceInteger programming is a thriving area of optimization, which is applied nowadays to a multitude of human endeavors, thanks to high quality software. It was developed over several decades and is still evolving rapidly.The goal of this book is to present the mathematical foundations of integer programming, with emphasis on the techniques that are most successful in current software implementations: convexification and enumeration.This textbook is intended for a graduate course in integer programming in M.S. or Ph.D. programs in applied mathematics, operations research, industrial engineering, or computer science.To better understand the excitement that is generated today by this area of mathematics, it is helpful to provide a historical perspective.Babylonian tablets show that mathematicians were already solving systems of lin...

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This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By "perfect formulation", we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequalities that is exponential in the size of the data needed to describe the problem. Here we are particularly interested in situations where the addition of a polynomial number of extra variables allows a formulation with a polynomial number of inequalities. Such formulations are called "compact extended formulations". We survey various tools for deriving and studying extended formulations, such as Fourier's procedure for projection, Minkowski-Weyl's theorem, Balas' theorem for the union of polyhedra, Yannakakis' theorem on the size of an extended formulation, dynamic programming, and variable discretization. For each tool that we introduce, we present one or several examples of how this tool is applied. In particular, we present compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set. We also present Bienstock's approximate compact extended formulation for the knapsack problem, Goemans' result on the size of an extended formulation for the permutahedron, and the Faenza-Kaibel extended formulation for orbitopes.

We consider a model that arises in integer programming and show that all irredundant inequalities are obtained from maximal lattice-free convex sets in an affine subspace. We also show that these sets are polyhedra. The latter result extends a theorem of Lovasz characterizing maximal lattice-free convex sets in R(n)

We present an algorithm that determines in polytime whether a graph contains an even hole. The algorithm is based on a decomposition theorem for even-hole-free graphs obtained in Part I of this work. We also give a polytime algorithm to find an even hole in a graph when one exists.

In Mathematical Programming 2003, Gomory and Johnson conjecture that the facets of the infinite group problem are always generated by piecewise linear functions. In this paper we give an example showing that the Gomory-Johnson conjecture is false.

We prove a decomposition theorem for even-hole-free graphs. The decompositions used are 2-joins and star, double-star and triple-star cutsets. This theorem is used in the second part of this paper to obtain a polytime recognition algorithm for even-hole-free graphs.

This survey is concerned with the size of perfect formulations for combinatorial optimization problems. By "perfect formulation", we mean a system of linear inequalities that describes the convex hull of feasible solutions, viewed as vectors. Natural perfect formulations often have a number of inequalities that is exponential in the size of the data needed to describe the problem. Here we are particularly interested in situations where the addition of a polynomial number of extra variables allows a formulation with a polynomial number of inequalities. Such formulations are called "compact extended formulations". We survey various tools for deriving and studying extended formulations, such as Fourier's procedure for projection, Minkowski-Weyl's theorem, Balas' theorem for the union of polyhedra, Yannakakis' theorem on the size of an extended formulation, dynamic programming, and variable discretization. For each tool that we introduce, we present one or several examples of how this tool is applied. In particular, we present compact extended formulations for several graph problems involving cuts, trees, cycles and matchings, and for the mixing set. We also present Bienstock's approximate compact extended formulation for the knapsack problem, Goemans' result on the size of an extended formulation for the permutahedron, and the Faenza-Kaibel extended formulation for orbitopes.