The purpose of this monograph is a coherent introduction to global methods in combinatorial optimization. By “global” we mean those based on morphisms, i.e. maps between instances of a problem which preserve the essential features of that problem. This approach has been systematically developed in algebra, starting with the work of Jordan in 1870 (see [90]). Lie's work on continuous groups, which he intended to apply to differential equations, and Klein's work on discrete groups and geometry (the Erlanger program) resulted from a trip the two made to Paris where they were exposed to Jordan's ideas. Global methods are inherent in all of mathematics, but the benefits of dealing with morphisms do not always justify the effort required and it has also been ignored in many areas. This has been especially true of combinatorics which is viewed by most of its practitioners as the study of finite mathematical structures, such as graphs, posets and designs, the focus being on problem-solving rather than theory-building.

What kinds of results can global methods lead to in combinatorics? Notions of symmetry, product decomposition and reduction abound in the combinatorial literature and these are by nature global concepts. Can we use the symmetry or product decomposition of a particular combinatorial problem to systematically reduce its size and complexity? Many of our results give positive answers to this question. We are not claiming, however, that the global point of view is the only valid one. On the contrary, we are endeavoring to show that global methods are complimentary to other approaches. Our focus is on global methods because they present opportunities which still remain largely unexploited.