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In mathematical programming and polyhedral combinatorics, the Hirsch conjecture stated that every graph of ad-dimensional polytope with $n$ facets has diameter no more than n &minus; d. It was proven by Francisco Santos Leal in 2010 that the conjecture is not true in general.

Counterexamples
every graph of an is the generally false statement that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n &minus; d. That is, any two vertices of the polytope must be connected to each other by a path of length at most n &minus; d. The conjecture was first put forth in a letter by Warren M. Hirsch to George B. Dantzig in 1957 and was motivated by the analysis of the simplex method in linear programming, as the diameter of a polytope provides a lower bound on the number of steps needed by the simplex method.

The Hirsch conjecture was proven for d < 4 and for various special cases. The best known upper bounds showed only that polytopes have sub-exponential diameter as a function of n and d. After more than fifty years, a counter-example was announced in May 2010 by Francisco Santos, from the University of Cantabria. The result was presented at the conference 100 Years in Seattle: the mathematics of Klee and Grünbaum and appeared in Annals of Mathematics. Specifically, the paper presented a 43-dimensional polytope of 86 facets with a diameter of more than 43. The counterexample has no direct consequences for the analysis of the simplex method, as it does not rule out the possibility of a larger but still linear or polynomial number of steps.

Various equivalent formulations of the problem had been given, such as the d-step conjecture, which states that the diameter of any 2d-facet polytope in d-dimensional Euclidean space is no more than d.