Abstract
Here is the story of how this paper was written. (a) Independently, Alan and Joe discovered this easy theorem: if the “right hand side” consists of integers, and if the matrix is “totally unimodular”, then the vertices of the polyhedron defined by the linear inequalities will all be integral. This is easy to prove and useful. As far as we know, this is the only part of our theorem that anyone has ever used.
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References
A.J. Hoffman, Some recent applications of the theory of linear inequalities to extremal combinatorial analysis, Proceedings Symposium Applied Mathematics 10, American Mathematical Society, 1960, pp. 113–128.
A.J. Hoffman, Total unimodularity and combinatorial theorems, Linear Algebra and its Applications 13 (1976) 103–108.
P.D. Seymour, Decomposition of regular matroids, Journal of Combinatorial Theory B 29 (1980) 305–359.
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Hoffman, A.J., Kruskal, J.B. (2010). Integral Boundary Points of Convex Polyhedra. In: Jünger, M., et al. 50 Years of Integer Programming 1958-2008. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-68279-0_3
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DOI: https://doi.org/10.1007/978-3-540-68279-0_3
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