In mathematical programming and polyhedral combinatorics, Hirsch's conjecture states that the edge-vertex graph of an n-facet polytope in d-dimensional Euclidean space has diameter no more than n − d. That is, any two vertices of the polytope must be connected to each other by a path of length at most n − d. The conjecture was first put forth in a letter by Warren M. Hirsch to George B. Dantzig in 1957[1][2] 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.
Hirsch's conjecture was proven for d < 4 and for various special cases,[3] The best known upper bounds showed only that polytopes have sub-exponential diameter as a function of n and d.[4] However, after more than fifty years, a counter-example was announced in May 2010 by Francisco Santos Leal, from the University of Cantabria.[5][6][7] The result was presented at the conference 100 Years in Seattle: the mathematics of Klee and Grünbaum. 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.[1][8] The d-step conjecture was known to be true for d < 6,[8] but the general case was also disproved when a counter-example was found, using a 43-dimensional polytope of 86 facets with a diameter of more than 43.[5] The announced counterexample would have no direct consequences for the analysis of the simplex method, as it would not rule out the possibility of a larger but still linear or polynomial number of steps.