In mathematical programming and polyhedral combinatorics, the Hirsch conjecture is the statement 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[es] 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 conjecture is now known to be false in general.
The Hirsch conjecture was proven for d < 4 and for various special cases, while the best known upper bounds on the diameter are only sub-exponential in n and d. After more than fifty years, a counter-example was announced in May 2010 by Francisco Santos Leal, 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.
- Dantzig, George B. (1963), Linear Programming and Extensions, Princeton Univ. Press. Reprinted in the series Princeton Landmarks in Mathematics, Princeton University Press, 1998.
- Kalai, Gil (10 May 2010). "Francisco Santos Disproves the Hirsch Conjecture". Retrieved 11 May 2010.
- Kalai, Gil; Kleitman, Daniel J. (1992), "A quasi-polynomial bound for the diameter of graphs of polyhedra", Bulletin of the American Mathematical Society, 26 (2): 315–316, arXiv: , doi:10.1090/S0273-0979-1992-00285-9, MR 1130448.
- Klee, Victor; Walkup, David W. (1967), "The d-step conjecture for polyhedra of dimension d < 6", Acta Mathematica, 133: 53–78, doi:10.1007/BF02395040, MR 0206823.
- Miranda, Eva (2012), "The Hirsch conjecture has been disproved: An interview with Francisco Santos" (PDF), Newsletter of the European Mathematical Society (86): 31–36.
- Naddef, Denis (1989), "The Hirsch conjecture is true for (0,1)-polytopes", Mathematical Programming, 45 (1): 109–110, doi:10.1007/BF01589099, MR 1017214.
- Santos, Francisco (2011), "A counterexample to the Hirsch conjecture", Annals of Mathematics, Princeton University and Institute for Advanced Study, 176 (1): 383–412, arXiv: , doi:10.4007/annals.2012.176.1.7, MR 2925387
- Ziegler, Günter M. (1994), "The Hirsch Conjecture", Lectures on Polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag, pp. 83–93.