Hirsch conjecture

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.

Notes

1. ^ ^{a b} Ziegler (1994), p. 84.
2. ^ Dantzig (1963), pp. 160 and 168.
3. ^ E.g. see Naddef (1989) for 0-1 polytopes.
4. ^ Kalai & Kleitman (1992).
5. ^ ^{a b} Santos (2010).
6. ^ Kalai (2010).
7. ^ http://gaussianos.com/francisco-santos-encuentra-un-contraejemplo-que-refuta-la-conjetura-de-hirsch/
8. ^ ^{a b} Klee & Walkup (1967).

References

• 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". http://gilkalai.wordpress.com/2010/05/10/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:math/9204233, doi:10.1090/S0273-0979-1992-00285-9, MR1130448 .
• 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, MR0206823 .
• Naddef, Denis (1989), "The Hirsch conjecture is true for (0,1)-polytopes", Mathematical Programming 45 (1): 109–110, doi:10.1007/BF01589099, MR1017214 .
• Santos, Francisco (2010). "A counterexample to the Hirsch conjecture". arXiv:1006.2814 [math.CO]. 
• Ziegler, Günter M. (1994), "The Hirsch Conjecture", Lectures on Polytopes, Graduate Texts in Mathematics, 152, Springer-Verlag, pp. 83–93 .