If you consider a polyhedron defined in d dimensions by the intersection of n half-spaces, then the Hirsch conjecture says that the diameter of the 1-skeleton of such a polyhedron is at most n-d. We've been using n dimensions (variables) and m half-spaces (constraints), but I will stick with the n,d notation --- you can do the translation. Kalai and Kleitman showed (in this two-page paper) that the diameter is at most n^{2 + log d} --- and this is essentially the best result known.

This problem was of interest to us, because if the diameter of some LP polytope were indeed large, then the simplex algorithm starting from a bad vertex might require a large number of steps. Of course, we only care about polynomiality, so we would be happy with the polynomial Hirsch conjecture: the diameter of the 1-skeleton is at most poly(n,d).

Some more resources:

- A series of blog posts by Gil Kalai's blog: here is one of them. His posts contain a huge wealth of information about the conjecture, its variants and generalizations.
- A recent survey by Santos and Kim.

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