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In 1958, Ralph E. Gomory transformed the field of integer programming when he published a short paper that described his cutting-plane algorithm for pure integer programs and announced that the method could be refined to give a finite algorithm for integer programming. In January of 2008, to commemorate the anniversary of Gomory's seminal paper, a special session celebrating fifty years of integer programming was held in Aussois, France, as part of the 12th Combinatorial Optimization Workshop. This book is based on the material presented during this session. 50 Years of Integer Programming offers an account of featured talks at the 2008 Aussois workshop, namely - Michele Conforti, Gérard Cornuéjols, and Giacomo Zambelli: Polyhedral Approaches to Mixed Integer Linear Programming - William Cook: 50+ Years of Combinatorial Integer Programming - Francois Vanderbeck and Laurence A. Wolsey: Reformulation and Decomposition of Integer Programs The book contains reprints of key historical articles together with new introductions and historical perspectives by the authors: Egon Balas, Michel Balinski, Jack Edmonds, Ralph E. Gomory, Arthur M. Geoffrion, Alan J. Hoffman & Joseph B. Kruskal, Richard M. Karp, Harold W. Kuhn, and Ailsa H. Land & Alison G. Doig. It also contains written versions of survey lectures on six of the hottest topics in the field by distinguished members of the integer programming community: - Friedrich Eisenbrand: Integer Programming and Algorithmic Geometry of Numbers - Raymond Hemmecke, Matthias Köppe, Jon Lee, and Robert Weismantel: Nonlinear Integer Programming - Andrea Lodi: Mixed Integer Programming Computation - Francois Margot: Symmetry in Integer Linear Programming - Franz Rendl: Semidefinite Relaxations for Integer Programming - Jean-Philippe P. Richard and Santanu S. Dey: The Group-Theoretic Approach to Mixed Integer Programming Integer programming holds great promise for the future, and continues to build on its foundations. Indeed, Gomory's finite cutting-plane method for the pure integer case is currently being reexamined and is showing new promise as a practical computational method. This book is a uniquely useful celebration of the past, present and future of this important and active field. Ideal for students and researchers in mathematics, computer science and operations research, it exposes mathematical optimization, in particular integer programming and combinatorial optimization, to a broad audience.

Combinatorial optimization -- Congresses. --- Combinatorial optimization. --- Integer programming -- Congresses. --- Integer programming. --- Integer programming --- Combinatorial optimization --- Civil & Environmental Engineering --- Mathematics --- Algebra --- Operations Research --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Mathematics. --- Operations research. --- Decision making. --- Computer science --- Mathematical optimization. --- Combinatorics. --- Optimization. --- Discrete Mathematics in Computer Science. --- Operation Research/Decision Theory. --- Programming (Mathematics) --- Mathematical optimization --- Combinatorics

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This volume constitutes the proceedings of the Fourth International Conference on Integer Programming and Combinatorial Optimization, IPCO '95, held in Copenhagen in May 1995 under the sponsorship of the Mathematical Programming Society. Integer programming and combinatorial optimization provide a fruitful theoretical and algorithmic basis for the solution of a number of optimization problems occuring in real-world situations, such as production planning and scheduling, routing, crew scheduling, or network construction. This volume presents 36 revised papers selected from a total of 105 submissions and offers a representative up-to-date snapshot on the state of the art in this interdisciplinary area of research and applications.

Integer programming --- Combinatorial optimization --- Programmation en nombres entiers --- Optimisation combinatoire --- Congresses. --- Congrès --- Congresses --- Congrès --- Information theory. --- Distribution (Probability theory. --- Mathematics. --- Computer software. --- Numerical analysis. --- Combinatorics. --- Theory of Computation. --- Probability Theory and Stochastic Processes. --- Applications of Mathematics. --- Algorithm Analysis and Problem Complexity. --- Numerical Analysis. --- Combinatorics --- Algebra --- Mathematical analysis --- Software, Computer --- Computer systems --- Math --- Science --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Communication theory --- Communication --- Cybernetics --- Integer programming - Congresses --- Combinatorial optimization - Congresses

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This volume contains the papers selected for presentation at IPCO VIII, the Eighth Conference on Integer Programming and Combinatorial Optimization, Utrecht, The Netherlands, 2001. This meeting isa forum for researchers and practitioners working on various aspects of integer programming and combi- torial optimization. The aim is to present recent developments in theory, com- tation, and application of integer programming and combinatorial optimization. Topics include, but are not limited to: approximation algorithms, branch and bound algorithms, computational biology, computational complexity, compu- tional geometry, cutting plane algorithms, diophantine equations, geometry of numbers, graph and network algorithms, integer programming, matroids and submodular functions, on-line algorithms, polyhedral combinatorics, scheduling theory and algorithms, and semide nit e programs. IPCO was established in 1988 when the rs t IPCO program committee was formed. The locations and years of the seven rs t IPCO conferences were: IPCO I, Waterloo (Canada) 1990, IPCO II, Pittsburgh (USA) 1992, IPCO III, - ice (Italy) 1993, IPCO IV, Copenhagen (Denmark) 1995, IPCO V, Vancouver (Canada) 1996, IPCO VI, Houston (USA) 1998, IPCO VII, Graz (Austria) 1999. IPCO is held every year in which no MPS (Mathematical Programming Society) International Symposium takes place. Since the MPS meeting is triennial, IPCO conferences are held twice in every three-year period. Asa rule, IPCO is held somewhere in Northern America in even years, and somewhere in Europe in odd years.

Integer programming --- Combinatorial optimization --- Operations Research --- Civil & Environmental Engineering --- Engineering & Applied Sciences --- Congresses --- Mathematics. --- Information technology. --- Business --- Algorithms. --- Computer science --- Probabilities. --- Combinatorics. --- Probability Theory and Stochastic Processes. --- Algorithm Analysis and Problem Complexity. --- Discrete Mathematics in Computer Science. --- IT in Business. --- Data processing. --- Programming (Mathematics) --- Distribution (Probability theory. --- Computer software. --- Computational complexity. --- Combinatorics --- Algebra --- Mathematical analysis --- IT (Information technology) --- Technology --- Telematics --- Information superhighway --- Knowledge management --- Complexity, Computational --- Electronic data processing --- Machine theory --- Software, Computer --- Computer systems --- Distribution functions --- Frequency distribution --- Characteristic functions --- Probabilities --- Computer science—Mathematics. --- Business—Data processing. --- Algorism --- Arithmetic --- Probability --- Statistical inference --- Combinations --- Mathematics --- Chance --- Least squares --- Mathematical statistics --- Risk --- Foundations --- Integer programming - Congresses --- Combinatorial optimization - Congresses

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Studies in integer programming

Numerical methods of optimisation --- Programming --- Integer programming --- 519.85 --- 681.3*F22 --- 681.3*G16 --- Programming (Mathematics) --- 681.3*G16 Optimization: constrained optimization; gradient methods; integer programming; least squares methods; linear programming; nonlinear programming (Numericalanalysis) --- Optimization: constrained optimization; gradient methods; integer programming; least squares methods; linear programming; nonlinear programming (Numericalanalysis) --- 519.85 Mathematical programming --- Mathematical programming --- 681.3*F22 Nonnumerical algorithms and problems: complexity of proof procedures; computations on discrete structures; geometrical problems and computations; pattern matching --See also {?681.3*E2-5}; {681.3*G2}; {?681.3*H2-3} --- Nonnumerical algorithms and problems: complexity of proof procedures; computations on discrete structures; geometrical problems and computations; pattern matching --See also {?681.3*E2-5}; {681.3*G2}; {?681.3*H2-3} --- Integer programming - Congresses

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The volume contains the papers selected for presentation at IPCO 2008, the 13th International Conference on Integer Programming and Combinatorial - timization that was held in Bertinoro (Italy), May 26–28, 2008. The IPCO series of conferences, sponsored by the Mathematical Progr- ming Society, highlights recent developments in theory, computation, and app- cation of integer programming and combinatorial optimization. The ?rst conf- ence took place in 1990; starting from IPCO 1995, the proceedings are published in the Lecture Notes in Computer Science series. The 12 previous IPCO conferences were held in Waterloo (Canada) 1990, Pittsburgh (USA) 1992, Erice (Italy) 1993, Copenhagen (Denmark) 1995 [LNCS 920], Vancouver (Canada) 1996 [LNCS 1084], Houston (USA) 1998 [LNCS 1412], Graz (Austria) 1999 [LNCS 1610], Utrecht (The Netherlands) 2001 [LNCS 2081], Boston (USA) 2002 [LNCS 2337], New York (USA) 2004 [LNCS 2986], Berlin (Germany) 2005 [LNCS 3509], and Ithaca (USA) 2007 [LNCS 4168]. The c- ference is not held in the years when the International Symposium of the Ma- ematical Programming Society takes place.

Integer programming --- Combinatorial optimization --- Programmation en nombres entiers --- Optimisation combinatoire --- Congresses. --- Congrès --- Operations Research --- Computer Science --- Engineering & Applied Sciences --- Civil & Environmental Engineering --- Congresses --- Computer science. --- Algorithms. --- Numerical analysis. --- Computer science --- Computer graphics. --- Computer Science. --- Discrete Mathematics in Computer Science. --- Numeric Computing. --- Algorithm Analysis and Problem Complexity. --- Computer Graphics. --- Mathematics. --- Automatic drafting --- Graphic data processing --- Graphics, Computer --- Computer art --- Graphic arts --- Electronic data processing --- Engineering graphics --- Image processing --- Computer mathematics --- Discrete mathematics --- Mathematical analysis --- Algorism --- Algebra --- Arithmetic --- Informatics --- Science --- Digital techniques --- Mathematics --- Foundations --- Programming (Mathematics) --- Computational complexity. --- Electronic data processing. --- Computer software. --- Software, Computer --- Computer systems --- ADP (Data processing) --- Automatic data processing --- Data processing --- EDP (Data processing) --- IDP (Data processing) --- Integrated data processing --- Computers --- Office practice --- Complexity, Computational --- Machine theory --- Automation --- Computer science—Mathematics. --- Integer programming - Congresses --- Combinatorial optimization - Congresses --- Discrete mathematics. --- Numerical Analysis. --- Discrete mathematical structures --- Mathematical structures, Discrete --- Structures, Discrete mathematical --- Numerical analysis