What is Combinatorics? What is Optimization? Department of Combinatorics and Optimization University of Waterloo, Ontario, Canada http://www.math.uwaterloo.ca 1. Introduction The department of C&O o¤ers programs in these two areas. These pages tells you a little bit about them. In industry, commerce, govern- ment, indeed in all walks of life, one frequently needs answers to ques- tions concerning operational e¢ciency. Thus an architect may need to know how to lay out a factory ‡oor so that the article being manufac- tured does not have to be moved about too much as it goes from one machine tool to another; the manager of a shipping company needs to plan the itineraries of his ships so as to increase the amount of goods handled, while avoiding costly waiting-around in docks. A telecom- munications engineer may want to know how best to transmit signals so as to minimize the possibility of error on reception. Further exam- ples of problems of this sort are provided by the planning of a railroad time-table to ensure that trains are available as and when needed, the synchronization of tra¢c lights, and many other real-life situations. Formerly such problems would usually be “solved” by imprecise methods giving results that were both unreliable and costly. Today, they are increasingly being subjected to rigid mathematical analysis, designed to provide methods for …nding exact solutions or highly reli- able estimates rather than vague approximations. Combinatorics and optimization provide many of the mathematical tools used for solving such problems. Combinatorics is the mathematics of discretely structured problems. Although its boundaries are not easily de…ned, combinatorics includes the theories of graphs, enumeration, designs and polyhedra. It is a subject which in the past was studied principally for its aesthetic ap- peal. Today’s modern technology with its vital concern for the dis- crete has given combinatorics new challenges and a new seriousness of purpose. In particular, since computers require discrete formula- tions of problems, combinatorics has become indispensable to modern computer science. 1 2 Optimization, or mathematical programming, is the study of maxi- mizing and minimizing functions subject to speci…ed boundary condi- tions or constraints. The functions to be optimized arise in engineer- ing, the physical and management sciences, and in various branches of mathematics. With the emergence of the computer age, optimization experienced a dramatic growth as a mathematical theory, enhancing both combinatorics and classical analysis. In its applications to en- gineering and management science, optimization forms an important part of the discipline of operations research. 2. Major Areas of Combinatorics and Optimization 2.1. Enumeration. Enumeration is concerned with determining the number of structures with prescribed properties, and is a frequently used tool in mathematics. Many enumeration problems arise from ranking and signi…cance testing in statistics, from probability theory, from telephone networks and from mathematics itself. Extensive cal- culations are often necessary, and the …eld of computational enumera- tion is an important one. The task of getting a computer to evaluate formulae obtained by applying enumerative methods is one requiring considerable experience both with computers and with combinatorics. 2.2. Combinatorial Designs. The study of designs deals with a very important and central problem of combinatorial theory, that of arrang- ing objects into patterns according to speci…ed rules. The principal mathematical tools employed are graph theory, number theory and linear and abstract algebra. The construction of magic squares is an example of a design problem, and before the start of the nineteenth century such problems were mainly of recreational interest. In more recent times the concept of a geometry involving only …nitely many points has been developed, generalized, and built into the elegant and useful theory of combinatorial designs. Today this area embodies the mathematical tools of such applied areas as the design of experiments, tournament scheduling, information retrieval and coding theory. 2.3. Graph Theory. Based upon the simple idea of points intercon- nected by lines, graph theory combines these basic ingredients into a rich and useful theory, which provides powerful tools for constructing models and solving problems concerning discrete arrangements of ob- jects. Technology today poses a great number of problems that require the construction of complex systems through speci…c arrangements of their components. These include problems in the scheduling of indus- trial processes, communication systems, electrical networks, organic- chemical identi…cation, economics and numerous other applied areas. 3 2.4. Linear Programming. Mathematical programming deals with the problem of determining optimal allocations of limited resources to meet given objectives. The resources may correspond to people, mate- rials, money, land or simply to abstract numerical quantities. Out of all permissible allocations of the resources, it is desired to …nd the one or ones which maximize or minimize some numerical quantity such as pro…t or cost. Linear programming deals with that class of program- ming problems for which the function to be optimized is linear and all relations among the variables corresponding to resources are linear. This problem was …rst formulated and solved in the late 1940s. Rarely has a new mathematical technique found such a wide range of practical applications and simultaneously received so thorough a theoretical development in such a short period of time. Today, this theory is being successfully applied to problems of capital budgeting, design of diets, conservation of resources, games of strategy, economic growth prediction and transportation systems. In very recent times linear programming theory has also helped resolve and unify many outstanding problems in combinatorics. 2.5. Nonlinear Optimization. Linear programming has proved an extremely powerful tool, both in modeling real-world problems and as a widely applicable mathematical theory. However, many interest- ing optimization problems are nonlinear. The study of such problems involves a diverse blend of linear algebra, multivariate calculus, numeri- cal analysis and computing techniques. Important special areas include the design of computational algorithms (including interior point tech- niques for linear programing), the geometry and analysis of convex sets and functions, and the study of specially structured problems such as quadratic programming. Nonlinear optimization provides fundamental insights into mathematical analysis, and is widely used in the applied sciences, in areas such as engineering design, regression analysis, inven- tory control, and geophysical exploration amont others. 2.6. Operations Research. The term Operations Research (OR for short) arose in the 1940’s when research was carried out on the design and analysis of mathematical models for military operations. Since that time the scope of OR has expanded to include economic, marketing and corporate planning problems. The growing complexity of management has necessitated the development of sophisticated mathematical tech- niques for planning and decision-making, and OR features prominently in this decision-making process by providing a quantitative evaluation of alternative policies, plans and decisions. The mathematical disci- plines most widely used in OR include mathematical programming, 4 probability and statistics, and computer science. Some areas of OR, such as inventory and production control, and scheduling theory, have grown into subdisciplines in their own right and have become largely indispensable in the modern world. 2.7. Combinatorial Optimization. Packing, covering and partition- ing, which are examples of integer programming problems, are the prin- cipal mathematical topics in the interface between combinatorics and optimization. Combinatorial optimization is the study of these prob- lems. It deals with the classi…cation of integer programming problems according to the complexity of known algorithms for solving them and with the design of good algorithms for solving special subclasses of them. In particular, problems of network ‡ows, matchings and their matroid generalizations are studied. This subject is one of the unify- ing elements of combinatorics, optimization, operations research and computer science. 3. Degree Programs The C&O Department isthehomeoftwoMajorprograms,theC&O Program and the OR Program, each leading to an Honours Bachelor of Mathematics degree. The curricula of these programs are especially appropriate for those showing special talent and interest in modern mathematics and who anticipate graduate studies in mathematics, op- erations research or computer science. The programs have been de- signed to achieve a balance between a strong core of third-year studies in algebra, analysis, combinatorics, and optimization, and more ad- vanced fourth-year studies in at least some of these areas. Students in the C&O Program are encouraged to broaden their mathematical backgrounds further by taking additional third and fourth year courses from at least one of the other four departments..