1 Solution of a Large-Scale Traveling-Salesman Problem... 7 George B. Dantzig, Delbert R. Fulkerson, and Selmer M. Johnson

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1 Part I The Early Years 1 Solution of a Large-Scale Traveling-Salesman Problem George B. Dantzig, Delbert R. Fulkerson, and Selmer M. Johnson 2 The Hungarian Method for the Assignment Problem Harold W. Kuhn 3 Integral Boundary Points of Convex Polyhedra Alan J. Hoffman and Joseph B. Kruskal 4 Outline of an Algorithm for Integer Solutions to Linear Programs and An Algorithm for the Mixed Integer Problem Ralph E. Gomory 5 An Automatic Method for Solving Discrete Programming Problems. 105 Ailsa H. Land and Alison G. Doig 6 Integer Programming: Methods, Uses, Computation Michel Balinski 7 Matroid Partition Jack Edmonds 8 Reducibility Among Combinatorial Problems Richard M. Karp 9 Lagrangian Relaxation for Integer Programming Arthur M. Geoffrion 10 Disjunctive Programming Egon Balas xv

2 xvi Part II From the Beginnings to the State-of-the-Art 11 Polyhedral Approaches to Mixed Integer Linear Programming Michele Conforti, Gérard Cornuéjols, and Giacomo Zambelli 11.1 Introduction Mixed integer linear programming Historical perspective Cutting plane methods Polyhedra and the fundamental theorem of integer programming Farkas lemma and linear programming duality Carathéodory s theorem The theorem of Minkowski-Weyl Projections The fundamental theorem for MILP Valid inequalities Facets Union of polyhedra Split disjunctions One-side splits, Chvátal inequalities Gomory s mixed-integer inequalities Equivalence of split closure and Gomory mixed integer closure Polyhedrality of closures The Chvátal closure of a pure integer set The split closure of a mixed integer set Lift-and-project Lift-and-project cuts Strengthened lift-and-project cuts Improving mixed integer Gomory cuts by lift-and-project Sequential convexification Rank Chvatal rank Split rank References Fifty-Plus Years of Combinatorial Integer Programming William Cook 12.1 Combinatorial integer programming The TSP in the 1950s Proving theorems with linear-programming duality Cutting-plane computation Jack Edmonds, polynomial-time algorithms, and polyhedral combinatorics Progress in the solution of the TSP Widening the field of application in the 1980s

3 xvii 12.8 Optimization Separation State of the art References Reformulation and Decomposition of Integer Programs François Vanderbeck and Laurence A. Wolsey 13.1 Introduction Polyhedra, reformulation and decomposition Introduction Polyhedra and reformulation Decomposition Price or constraint decomposition Lagrangean relaxation and the Lagrangean dual Dantzig-Wolfe reformulations Solving the Dantzig-Wolfe relaxation by column generation Alternative methods for solving the Lagrangean dual Optimal integer solutions: branch-and-price Practical aspects Resource or variable decomposition Benders reformulation Benders with integer subproblems Block diagonal structure Computational aspects Extended formulations: problem specific approaches Using compact extended formulations Variable splitting I: multi-commodity extended formulations Variable splitting II Reformulations based on dynamic programming The union of polyhedra From polyhedra and separation to extended formulations Miscellaneous reformulations Existence of polynomial size extended formulations Hybrid algorithms and stronger dual bounds Lagrangean decomposition or price-and-price Cut-and-price Notes Polyhedra Dantzig-Wolfe and price decomposition Resource decomposition Extended formulations Hybrid algorithms and stronger dual bounds References

4 xviii Part III Current Topics 14 Integer Programming and Algorithmic Geometry of Numbers Friedrich Eisenbrand 14.1 Lattices, integer programming and the geometry of numbers Informal introduction to basis reduction The Hermite normal form Minkowski s theorem The LLL algorithm Kannan s shortest vector algorithm A randomized simply exponential algorithm for shortest vector Integer programming in fixed dimension The integer linear optimization problem Diophantine approximation and strongly polynomial algorithms Parametric integer programming References Nonlinear Integer Programming Raymond Hemmecke, Matthias Köppe, Jon Lee, and Robert Weismantel 15.1 Overview Convex integer maximization Fixed dimension Boundary cases of complexity Reduction to linear integer programming Convex integer minimization Fixed dimension Boundary cases of complexity Practical algorithms Polynomial optimization Fixed dimension and linear constraints: An FPTAS Semi-algebraic sets and SOS programming Quadratic functions Global optimization Spatial Branch-and-Bound Boundary cases of complexity Conclusions References Mixed Integer Programming Computation Andrea Lodi 16.1 Introduction MIP evolution A performance perspective A modeling/application perspective MIP challenges

5 xix A performance perspective A modeling perspective Conclusions References Symmetry in Integer Linear Programming François Margot 17.1 Introduction Preliminaries Detecting symmetries Perturbation Fixing variables Symmetric polyhedra and related topics Partitioning problems Dantzig-Wolfe decomposition Partitioning orbitope Asymmetric representatives Symmetry breaking inequalities Dynamic symmetry breaking inequalities Static symmetry breaking inequalities Pruning the enumeration tree Pruning with a fixed order on the variables Pruning without a fixed order of the variables Group representation and operations Enumerating all non-isomorphic solutions Furthering the reach of isomorphism pruning Choice of formulation Exploiting additional symmetries References Semidefinite Relaxations for Integer Programming Franz Rendl 18.1 Introduction Basics on semidefinite optimization Modeling with semidefinite programs Quadratic 0/1 optimization Max-Cut and graph bisection Stable sets, cliques and the Lovász theta function Chromatic number General graph partition Generic cutting planes SDP, eigenvalues and the Hoffman-Wielandt inequality The theoretical power of SDP Hyperplane rounding for Max-Cut Coloring Solving SDP in practice

6 xx Interior point algorithms Partial Lagrangian and the bundle method The spectral bundle method SDP and beyond Copositive and completely positive matrices Copositive relaxations References The Group-Theoretic Approach in Mixed Integer Programming Jean-Philippe P. Richard and Santanu S. Dey 19.1 Introduction The corner relaxation Linear programming relaxations Motivating example Gomory s corner relaxation Group relaxations: optimal solutions and structure Optimizing linear functions over the corner relaxation Using corner relaxations to solve MIPs Extended group relaxations Master group relaxations: definitions and inequalities Groups Master group relaxations of mixed integer programs A hierarchy of inequalities for master group problems Extreme inequalities Extreme inequalities of finite master group problems Extreme inequalities for infinite group problems A compendium of known extreme inequalities for finite and infinite group problems On the strength of group cuts and the group approach Absolute strength of group relaxation Relative strength of different families of group cuts Summary on strength of group cuts Conclusion and perspectives References Part IV DVD-Video / DVD-ROM

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