chapter 3 integral boundary points of convex polyhedragasarch/blogpapers/kruskalhoffman.pdf ·...

Chapter 3Integral Boundary Points of Convex Polyhedra

Alan J. Hoffman and Joseph B. Kruskal

Introduction by Alan J. Hoffman and Joseph B. Kruskal

Here is the story of how this paper was written.

(a) Independently, Alan and Joe discovered this easy theorem: if the “right handside” consists of integers, and if the matrix is “totally unimodular”, then the verticesof the polyhedron defined by the linear inequalities will all be integral. This is easyto prove and useful. As far as we know, this is the only part of our theorem thatanyone has ever used.

(b) But this was so easy, we each wanted to generalize it. Independently weworked hard to understand the cases where there are no vertices, i.e., the lowestdimensional faces of the polyhedron are 1-dimensional or higher. This was hard towrite and hard to read.

(c) At this point, Alan benefitted greatly from simplifications suggested by DavidGale and anonymous referees, but it was still not so simple.

(d) Independently, we both wondered: If the vertices were integral for every in-tegral right hand side, did this mean the matrix was totally unimodular? This isdiscussed in our paper, and also in References [1] and [2], especially the latter.

Harold Kuhn and Al Tucker saw drafts from both of us and realized that we wereworking on the same problem, so they suggested that we start working together, andthat Alan should send his latest draft to Joe to take the next step. For Joe, this turnedout to be the most exciting collaboration he had ever experienced; and he still feelsthat way today.

Alan J. HoffmanIBM Research, Yorktown Heights, New York, USAe-mail: [email protected]

Joseph B. KruskalBell Laboratories, Murray Hill, New Jersey, USAe-mail: [email protected]

49 M. Jünger et al. (eds.), 50 Years of Integer Programming 1958-2008, DOI 10.1007/978-3-540-68279-0_3, © Springer-Verlag Berlin Heidelberg 2010

50 Alan J. Hoffman and Joseph B. Kruskal

We must have met casually before the collaboration, but we never saw eachother during it nor for a long time afterwards. Joe knew nothing about Alan’s work.However Alan, who was working for the Navy, knew something about Joe’s workthrough a Navy report on a real operations research project on which Joe and BobAumann had gotten impressive results. (Many years later, in 2005, Bob won theNobel Prize for Economics.)

As it turned out, Joe merged our two papers—but did much more; Alan’s ideaswere very stimulating. When Alan got that version, he was also stimulated and madesubstantial improvements. Then Joe made further improvements, and finally Alandid the same. We could probably have made much more progress, but the deadlinefor publication cut off further work.

(e) One of our discoveries when collaborating was a new general class of totallyunimodular matrices . . . but several years later we were chagrined to learn from JackEdmonds that in the 1800’s Gustav Kirchoff (who was the inventor of Kirchoff’sLaws) had constructed a class of totally unimodular matrices of which ours wasonly a special case.

(f) The term “totally unimodular” is due to Claude Berge, and far superior to ourwishywashy phrase “matrices with the unimodular property”. Claude had a flair forlanguage.

(g) We had no thought about computational questions, practical or theoretical,that could be influenced by our work. We also did not imagine the host of interest-ing concepts, like total dual integrality, lattice polyhedral, etc. that would emerge,extending our idea. And we never dreamed that totally unimodular matrices couldbe completely described, see [3], because we didn’t anticipate that a mathematicianwith the great talent of Paul Seymour would get interested in these concepts.

After we wrote the paper, we met once in a while (a theater in London, a meetingin Washington), but our interests diverged and we never got together again pro-fessionally. The last time we met was almost 15 years ago, when Vasek Chvatalorganized at Rutgers a surprise 70th birthday party cum symposium for Alan. Joespoke about this paper and read some of the letters we wrote each other, includingreciprocal requests that each of us made begging the partner to forgive his stupidity.

References

1. A.J. Hoffman, Some recent applications of the theory of linear inequalities to extremal combi-natorial analysis, Proceedings Symposium Applied Mathematics 10, American MathematicalSociety, 1960, pp. 113–128.

2. A.J. Hoffman, Total unimodularity and combinatorial theorems, Linear Algebra and its Appli-cations 13 (1976) 103–108.

3. P.D. Seymour, Decomposition of regular matroids, Journal of Combinatorial Theory B 29(1980) 305–359.

3 Integral Boundary Points of Convex Polyhedra 51


The following article originally appeared as:

A.J. Hoffman and J.B. Kruskal, Integral Boundary Points of Convex Polyhedra, Lin-ear Inequalities and Related Systems (H.W. Kuhn and A.J. Tucker, eds.), PrincetonUniversity Press, 1956, pp. 223–246.

Copyright c© 1956 Princeton University Press, 1984 renewed PUP.

Reprinted by permission from Princeton University Press.

chapter 3 integral boundary points of convex polyhedragasarch/blogpapers/kruskalhoffman.pdf ·...

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