The atoms of integer programming

After thirty years of doing other things, I am once again doing some research on integer programming. It has been interesting and exciting for me to see what has changed during those thirty years. The practical side of integer programming has developed far more than I would have ever anticipated. Problems of a size and complexity that we would never have thought of attempting thirty years ago are being done routinely. And this has been accomplished not by some theoretical or conceptual breakthrough (the branch and cut methods generally used are fundamentally the same as before) but by the intelligent and persistent use of empirical methods and empirical learning. Added to this, of course, is 30 years of progress in electronics, which is worth another improvement factor of roughly 1000 in performance. On the theoretical side there is continued interest in special problems: the traveling salesman problem, for example, is very much alive and well, as are many graph theory problems. There is a large gap between the work on special problems and the work on large practical problems. The type of discourse in these two areas is quite different. One area deals in large test problems and how they run, the other is much more theoretical.