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By Paul C. Kettler


Warren M. Hirsch posed the conjecture which bears his name in a letter of 1957 to George B. Dantzig. Simply stated in geometric terms, Hirsch proposed that a convex polyhedron in dimension d with n facets admits a path of at most (n − d) edges connecting any two vertices. Hirsch posed his conjecture in the context of a linear program in d variables and n constraints as requiring no more than (n − d) pivots — steps of the simplex algorithm — on the shortest path to achieve an optimum. While the conjecture is known to be false for some unbounded polyhedra, over the years it has attracted much research attention for polytopes, and has been proved in special cases. This article contributes a general proof for polytopes

Year: 2009
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