We find in the literature three distinct connections between combinatorial

isoperimetric problems and partially ordered sets:

1. The reduction of edge and vertex-isoperimetric problems on graphs (EIP and VIP) to maximum weight ideal (MWI) problems on the compressibility or stability order (see Chapters 3, 4, 5 and 6).

2. J. B. Kruskal's observation, in 1969 [66], that a graph may be looked upon as a two-dimensional complex and then its (incident) EIP has a natural extension to arbitrary complexes (hypergraphs). The extension is called the minimum shadow problem (MSP). Kruskal had already solved the MSP [65] for the simplex in all dimensions, a result discovered independently by G. O. H. Katona [58]. The Kruskal–Katona theorem is probably the most widely known and applied of all combinatorial isoperimetric theorems. Kruskal went on in [66] to conjecture that our solution of the EIP on Qd, the graph of the d-cube (see Chapters 1 and 3), could be extended to the MSP on the complex of faces of the d-cube. He also suggested looking for more such analogs of the Kruskal–Katona theorem.

3. Scheduling problems are standard fare in applied combinatorial optimization. If the steps of a manufacturing process must be carried out in some serial order subject to given precedence constraints, and we wish to order the steps so as to minimize some functional of the ordering, such as the average time between when a step is completed and its last successor is completed, then we have a scheduling problem.

Why infinite graphs? The EIP, or any of its variants, would not seem suited to infinite graphs. On finite graphs we can always find a solution by brute force, evaluating |Θ(S)| for all 2|V| subsets of vertices. Even so, the finite problem is NP-complete, an analog of undecidability, and on infinite graphs it is very likely undecidable. Certainly there is no apparent solution.

The primary motivation for considering the EIP on infinite graphs is to develop global methods. Problems are the life blood of mathematics and there are some very large, i.e. finite but for all practical purposes infinite, graphs for which we would like to solve the EIP. The 120-cell, an exceptional regular solid in four dimensions, is the only regular solid for which we have not solved the EIP. It has 600 vertices so we prefer to call it the 600-vertex, V600. Another is the graph of the n-permutohedron, n ≥ 4, which has n! vertices. Solving those problems will require developing better methods than we have now. The regular tessellations of Euclidean space are relatively easy to work with but present some of the same kinds of technical problems as those higher dimensional semiregular and exceptional regular solids.

There are also problems arising in applications which bring us to consider isoperimetric problems on infinite graphs. The original application, solving a kind of layout problem if G is regarded as representing an electronic circuit, did not seem to make sense if G is infinite.

Almost forty years ago I was persuaded that combinatorial isoperimetric problems were worthy of systematic investigation. The edge- and vertexisoperimetric problems were clearly fundamental aspects of graph theory. They had already been applied to the wirelength and bandwidth problems on d-cubes and other graphs which had engineering implications. As analogs of the classical isoperimetric problem of Greek geometry they seemed certain to lead to further useful results. Over the years this analogy, with the pressure of prospective applications, has produced profound solution methods; spectral, global and variational.

It has been very difficult to bring closure to the writing of this monograph since every time I go over the material, new insights appear and demand to be included. Also, tempting new problems keep arising in science, engineering and mathematics itself. For instance, Lubotzky's monograph [75] has a whole chapter of unsolved problems. It seems certain that the subject will continue to progress for the foreseeable future, but life is short and we cannot wait until every significant question has been answered. Last week, in a conversation with T. H. Payne, colleague, collaborator and for many years a most reliable source of information about trends in computer science, I mentioned recent work on the profile scheduling problem (see Chapter 8). “Oh, yes,” he said with enthusiasm, “that has been applied to optimizing straight-line programs! A ‘live variable’ must be stored in a register, so the profile equals total storage time. But the latest thing is to minimize register width, the maximum number of registers required by a program.”

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.