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The book provides a unified, insightful, and modern treatment of the theory of integer optimization.

The book is used in the doctoral level course, "Integer and Combinatorial Optimization" at the Massachusetts Institute of Technology.

For solutions to exercises and other instructor resources, please contact Dimitris Bertsimas (dbertsim@mit.edu).

The chapters of the book are logically organized in four parts:

Part I: Formulations and relaxations includes Chapters 1-5 and discusses how to formulate integer optimization problems, how to enhance the formulations to improve the quality of relaxations, how to obtain ideal formulations, the duality of integer optimization and how to solve the resulting relaxations both practically and theoretically.

Part II: Algebra and geometry of integer optimization includes Chapters 6-8 and develops the theory of lattices, oulines ideas from algebraic geometry that have had an impact on integer optimization, and most importantly discusses the geometry of integer optimization, a key feature of the book. These chapters provide the building blocks for developing algorithms.

Part III: Algorithms for integer optimization includes Chapters 9-12 and develops cutting plane methods, integral basis methods, enumerative and heuristic methods and approximation algorithms. The key characteristic of our treatment is that our development of the algorithms is naturally based on the algebraic and geometric developments of Part II.

Part IV: Extensions of integer optimization includes Chapters 13 and 14, and treats mixed integer optimization and robust discrete optimization. Both areas are practically significant as real world problems have very often both continuous and discrete variables and have elements of uncertainty that need to be addressed in a tractable manner.

Distinguishing Characteristics Of This Book:

* Develops the theory of integer optimization from a new geometric perspective via integral generating sets;

* Emphasizes strong formulations, ways to improve them, integral polyhedra, duality, and relaxations;

* Discusses applications of lattices and algebraic geometry to integer optimization, including Grobner bases, optimization over polynomials and counting integer points in polyhedra;

* Contains a unified geometric treatment of cutting plane and integral basis methods;

* Covers enumerative and heuristic methods, including local search over exponential neighborhoods and simulated annealing;

* Presents the major methods to construct approximation algorithms: primal-dual, randomized rounding, semidefinite and enumerative methods;

* Provides a unified treatment of mixed integer and robust discrete optimization;

* Includes a large number of examples and exercises developed through extensive classroom use..
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How to select and apply the right paint or coating for any surface The first major reference to help you choose the correct paint or other finish to do the job best on a particular surface exposed to a particular environment. Experts in the field give full advice on testing surface preparation, application, corrosion prevention, and troubleshooting. The handbook covers wood, metal, composites, and masonry, as well as marine applications and roof coatings. A ``must'' working tool for contractors, architects, engineers, specification writers, and paint dealers..
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The concept of "community" is ubiquitous in the way we talk and think about life in the twentyfirst century. Political and economic projects from rainforest conservation to urban empowerment zones focus on "the community" as the appropriate vehicle and target of change. Some scholars see a decline of community and predict dire social consequences; others criticize the concept itself for its ideological baggage and lack of clear definition. Moving the debate to a deeper level, the contributors to this volume aspire to understand the various ways "community" is deployed and the work it performs in different contexts. They compare the many cases where scholars and activists use "community" generically with instances in which the notion of community is less pervasive or even non-existent. How does a community facilitate governance or capital accumulation? In what ways does it articulate these two forces in local and translocal contexts? What are the unintended consequences of deploying the concept--and what, too, are the potential consequences of criticizing our fascination with it? The essays demonstrate the critical value of using community as the focus of analysis, rather than simply an empty category of heuristic or descriptive convenience..
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We are what we eat: our food defines us as individual women and men, as families and communities, and as members of our race, our class, and our nation. In this book, Mary Weismantel uses four different facets of the social life of food--diet, cuisine, discourse, and practice--to draw a richly detailed and compelling portrait of one South American community during the 1980s. The foods eaten in Zumbagua, an indigenous parish of highland Ecuador, are key to understanding what holds this distinctive people together in the face of tremendous economic and cultural challenges, as well as what divides them. The detailed discussion of diet is surprisingly revealing. Ancient histories emerge from the origins of staple crops like barley and potatoes, while recent trends, such as the substitution of purchased candies and colas for too-expensive fruits and vegetables, expose an ongoing ecological and economic crisis. In her discussion of cuisine--the cultural rules by which foods become meals--Weismantel shows how the everyday work of women preparing food transforms a mundane physical necessity, into a deeply meaningful symbolic act. Differences between local and national cultures, everyday and special occasions, men and women, adults and children, family and friends are only some of the cultural messages transmitted through snacks and means. Further, this culinary language is a highly expressive political idiom. By analyzing conversations and arguments about food, this book shows how an apparently apolitical community engaged in agonized debates about survival in the face of endemic racism and accelerating poverty. Cooking oil and wild mustard, bread and gruel, white rice and brown barley all appear as highly charged symbols of assimilation or resistance. Lastly, the book moves into the kitchen itself, where kinship, generation and gender shape--and are shaped by--the practical work of feeding the family. Social changes, such as the feminization of agriculture, continually alter labor demands within and outside of the kitchen, creating new tensions and conflicts within the family. By retaining close attention to the food itself as it is prepared and consumed, this book explores these intimate family issues without ever losing sight of the larger forces involved. The kitchen stove is a final nexus between production, exchange, and consumption. In the end, the delicate balance between the labor and products that go out of the house, and the goods that come back in, determines economic survival. And it is by choosing what to allow in and what to exclude, and how to shape the finished product for their own consumption, that the people of Zumbagua exert a precarious cultural autonomy in the face of daunting difficulties. This book is both a richly specific document of their lives, and a significant theoretical statement about the anthropology of food..
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Throughout the world, the kitchen is the heart of family and community life. Yet, while everyone has a story to tell about their grandmother's kitchen, the myriad activities that go on in this usually female world are often devalued, and little scholarly attention has been paid to this crucial space in which family, gender, and community relations are forged and maintained. To give the kitchen the prominence and respect it merits, Maria Elisa Christie here offers a pioneering ethnography of kitchenspace in three central Mexican communities, Xochimilco, Ocotepec, and Tetecala.

Christie coined the term "kitchenspace" to encompass both the inside kitchen area in which everyday meals for the family are made and the larger outside cooking area in which elaborate meals for community fiestas are prepared by many women working together. She explores how both kinds of meal preparation create bonds among family and community members. In particular, she shows how women's work in preparing food for fiestas gives women status in their communities and creates social networks of reciprocal obligation. In a culture rigidly stratified by gender, Christie concludes, kitchenspace gives women a source of power and a place in which to transmit the traditions and beliefs of older generations through quasi-sacramental food rites.

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The chapters of this Handbook volume covers nine main topics that are representative of recent
theoretical and algorithmic developments in the field. In addition to the nine papers that present the state of the art, there is an article on
the early history of the field.

The handbook will be a useful reference to experts in the field as well as students and others who want to learn about discrete optimization.

All of the chapters in this handbook are written by authors who have made significant original contributions to their topics. Herewith a brief introduction to the chapters of the handbook.

"On the history of combinatorial optimization (until 1960)" goes back to work of Monge in the 18th century on the assignment problem and presents six problem areas: assignment, transportation,
maximum flow, shortest tree, shortest path and traveling salesman.

The branch-and-cut algorithm of integer programming is the computational workhorse of discrete optimization. It provides the tools that have been implemented in commercial software such as CPLEX
and Xpress MP that make it possible to solve practical problems in supply chain, manufacturing, telecommunications and many other areas.
"Computational integer programming and cutting planes" presents the key ingredients
of these algorithms.

Although branch-and-cut based on linear programming relaxation is the most widely used integer programming algorithm, other approaches are
needed to solve instances for which branch-and-cut performs poorly and to understand better the structure of integral polyhedra. The next three chapters discuss alternative approaches.

"The structure of group relaxations" studies a family of polyhedra obtained by dropping certain
nonnegativity restrictions on integer programming problems.

Although integer programming is NP-hard in general, it is polynomially solvable in fixed dimension. "Integer programming, lattices, and results in fixed dimension" presents results in this area including algorithms that use reduced bases of integer lattices that are capable of solving certain classes of integer programs that defy solution by branch-and-cut.

Relaxation or dual methods, such as cutting plane algorithms,progressively remove infeasibility while maintaining optimality to the relaxed problem. Such algorithms have the disadvantage of
possibly obtaining feasibility only when the algorithm terminates.Primal methods for integer programs, which move from a feasible solution to a better feasible solution, were studied in the 1960's
but did not appear to be competitive with dual methods. However,recent development in primal methods presented in "Primal integer programming" indicate that this approach is not just interesting theoretically but may have practical implications as well.

The study of matrices that yield integral polyhedra has a long tradition in integer programming. A major breakthrough occurred in the 1990's with the development of polyhedral and structural results
and recognition algorithms for balanced matrices. "Balanced matrices" is a tutorial on the
subject.

Submodular function minimization generalizes some linear combinatorial optimization problems such as minimum cut and is one of the fundamental problems of the field that is solvable in polynomial
time. "Submodular function minimization" presents the theory and algorithms of this subject.

In the search for tighter relaxations of combinatorial optimization problems, semidefinite programming provides a generalization of
linear programming that can give better approximations and is still polynomially solvable. This subject is discussed in "Semidefinite programming and integer programming".

Many real world problems have uncertain data that is known only probabilistically. Stochastic programming treats this topic, but until recently it was limited, for computational reasons, to
stochastic linear programs. Stochastic integer programming is now a high profile research area and recent developments are presented in
"Algorithms for stochastic mixed-integer programming
models".

Resource constrained scheduling is an example of a class of combinatorial optimization problems that is not naturally formulated with linear constraints so that linear programming based methods do
not work well. "Constraint programming" presents an alternative enumerative approach that is complementary to branch-and-cut. Constraint programming,primarily designed for feasibility problems, does not use a relaxation to obtain bounds. Instead nodes of the search tree are
pruned by constraint propagation, which tightens bounds on variables until their values are fixed or their domains are shown to be empty..
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