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.