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An Update on the Hirsch Conjecture


The Hirsch conjecture was posed in 1957 in a question from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than nd. The number n of facets is the minimum number of closed half-spaces needed to form the polytope and the conjecture asserts that one can go from any vertex to any other vertex using at most nd edges.

Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound nd is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.


  1. 1.

    Altshuler, A.: The Mani-Walkup spherical counterexamples to the W v -path conjecture are not polytopal. Math. Oper. Res. 10(1), 158–159 (1985)

  2. 2.

    Altshuler, A., Bokowski, J., Steinberg, L.: The classification of simplicial 3-spheres with nine vertices into polytopes and non-polytopes. Discrete Math. 31, 115–124 (1980)

  3. 3.

    Balinski, M.L.: The Hirsch conjecture for dual transportation polyhedra. Math. Oper. Res. 9(4), 629–633 (1984)

  4. 4.

    Barnette, D.: W v paths on 3-polytopes. J. Comb. Theory 7, 62–70 (1969)

  5. 5.

    Beck, M., Robins, S.: Computing the Continuous Discretely: Integer-point Enumeration in Polyhedra. Undergraduate Texts in Mathematics. Springer, Berlin (2007)

  6. 6.

    Björner, A., Brenti, F.: Combinatorics of Coxeter Groups. Graduate Texts in Mathematics, vol. 231. Springer, Berlin (2005)

  7. 7.

    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, Berlin (1997)

  8. 8.

    Borgwardt, K.H.: The average number of steps required by the simplex method is polynomial. Z. Oper. Res. 26, 157–177 (1982)

  9. 9.

    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

  10. 10.

    Bremner, D., Deza, A., Hua, W., Schewe, L.: More bounds on the diameters of convex polytopes. Preprint, 8 p. (2009). arXiv:0911.4982v1

  11. 11.

    Bremner, D., Schewe, L.: Edge-graph diameter bounds for convex polytopes with few facets. Preprint, 9 p. (2008). arXiv:0809.0915v3

  12. 12.

    Brightwell, G., van den Heuvel, J., Stougie, L.: A linear bound on the diameter of the transportation polytope. Combinatorica 26(2), 133–139 (2006)

  13. 13.

    Cunningham, W.H.: Theoretical properties of the network simplex method. Math. Oper. Res. 4, 196–208 (1979)

  14. 14.

    Dantzig, G.B.: Linear Programming and Extensions. Princeton University Press, Princeton (1963)

  15. 15.

    De Loera, J.A., Kim, E.D., Onn, S., Santos, F.: Graphs of transportation polytopes. J. Comb. Theory Ser. A 116(8), 1306–1325 (2009)

  16. 16.

    De Loera, J.A.: The many aspects of counting lattice points in polytopes. Math. Semesterber. 52(2), 175–195 (2005)

  17. 17.

    De Loera, J.A., Onn, S.: All rational polytopes are transportation polytopes and all polytopal integer sets are contingency tables. In: Proc. 10th Ann. Math. Prog. Soc. Symp. Integ. Prog. Combin. Optim. (Columbia University, New York, NY, June 2004). Lec. Not. Comp. Sci., vol. 3064, pp. 338–351. Springer, New York (2004)

  18. 18.

    De Loera, J.A., Rambau, J., Santos, F.: Triangulations: Structures for Algorithms and Applications. Algorithms Comput. Math. vol. 25 (to appear), Springer-Verlag

  19. 19.

    Dedieu, J.-P., Malajovich, G., Shub, M.: On the curvature of the central path of linear programming theory. Found. Comput. Math. 5, 145–171 (2005)

  20. 20.

    Deza, A., Terlaky, T., Zinchenko, Y.: Central path curvature and iteration-complexity for redundant Klee-Minty cubes. Adv. Mech. Math. 17, 223–256 (2009)

  21. 21.

    Deza, A., Terlaky, T., Zinchenko, Y.: A continuous d-step conjecture for polytopes. Discrete Comput. Geom. 41, 318–327 (2009)

  22. 22.

    Deza, A., Terlaky, T., Zinchenko, Y.: Polytopes and arrangements: diameter and curvature. Oper. Res. Lett. 36(2), 215–222 (2008)

  23. 23.

    Dyer, M., Frieze, A.: Random walks, totally unimodular matrices, and a randomised dual simplex algorithm. Math. Program. 64, 1–16 (1994)

  24. 24.

    Eisenbrand, F., Hähnle, N., Razborov, A., Rothvoß, T.: Diameter of polyhedra: limits of abstraction. Preprint, available at http://people.cs.uchicago.edu/~razborov/research.html. A preliminary version appeared in: Proceedings of the 25th Annual ACM Symposium on Computational Geometry (SoCG’09), ACM, New York (2009)

  25. 25.

    Fomin, S., Zelevinsky, A.: Y-systems and generalized associahedra. Ann. Math. 158(2), 977–1018 (2003)

  26. 26.

    Fritzsche, K., Holt, F.B.: More polytopes meeting the conjectured Hirsch bound. Discrete Math. 205, 77–84 (1999)

  27. 27.

    Gawrilow, E., Joswig, M.: Polymake: a framework for analyzing convex polytopes. In: Polytopes—combinatorics and computation (Oberwolfach, 1997). DMV Sem., vol. 29, pp. 43–73. Birkhäuser, Basel (2000)

  28. 28.

    Goldfarb, D., Hao, J.: Polynomial simplex algorithms for the minimum cost network flow problem. Algorithmica 8, 145–160 (1992)

  29. 29.

    Goodey, P.R.: Some upper bounds for the diameters of convex polytopes. Isr. J. Math. 11, 380–385 (1972)

  30. 30.

    Hačijan, L.G.: A polynomial algorithm in linear programming. Dokl. Akad. Nauk SSSR 244(5), 1093–1096 (1979) (in Russian)

  31. 31.

    Holt, F.B.: Blending simple polytopes at faces. Discrete Math. 285, 141–150 (2004)

  32. 32.

    Holt, F., Klee, V.: Many polytopes meeting the conjectured Hirsch bound. Discrete Comput. Geom. 20, 1–17 (1998)

  33. 33.

    Hurkens, C.: Personal communication (2007)

  34. 34.

    Kalai, G.: A subexponential randomized simplex algorithm. In: Proceedings of the 24th Annual ACM Symposium on the Theory of Computing, pp. 475–482. ACM, New York (1992)

  35. 35.

    Kalai, G.: Online blog http://gilkalai.wordpress.com. See, for example, http://gilkalai.wordpress.com/2008/12/01/a-diameter-problem-7/, December 2008

  36. 36.

    Kalai, G., Kleitman, D.J.: A quasi-polynomial bound for the diameter of graphs of polyhedra. Bull. Am. Math. Soc. 26, 315–316 (1992)

  37. 37.

    Karmarkar, N.: A new polynomial time algorithm for linear programming. Combinatorica 4(4), 373–395 (1984)

  38. 38.

    Kim, E.D., Santos, F.: Companion to “An update on the Hirsch conjecture”. Preprint, 22 p. (2009). arXiv:0912.4235v1

  39. 39.

    Klee, V.: Paths on polyhedra II. Pac. J. Math. 17(2), 249–262 (1966)

  40. 40.

    Klee, V., Kleinschmidt, P.: The d-step conjecture and its relatives. Math. Oper. Res. 12(4), 718–755 (1987)

  41. 41.

    Klee, V., Minty, G.J.: How good is the simplex algorithm? In: Inequalities, III: Proc. Third Sympos., Univ. California, Los Angeles, Calif., 1969; dedicated to the memory of Theodore S. Motzkin, pp. 159–175. Academic Press, New York (1972)

  42. 42.

    Klee, V., Walkup, D.W.: The d-step conjecture for polyhedra of dimension d<6. Acta Math. 133, 53–78 (1967)

  43. 43.

    Kleinschmidt, P., Onn, S.: On the diameter of convex polytopes. Discrete Math. 102(1), 75–77 (1992)

  44. 44.

    Larman, D.G.: Paths of polytopes. Proc. Lond. Math. Soc. 20(3), 161–178 (1970)

  45. 45.

    Mani, P., Walkup, D.W.: A 3-sphere counterexample to the W v -path conjecture. Math. Oper. Res. 5(4), 595–598 (1980)

  46. 46.

    Matoušek, J., Sharir, M., Welzl, E.: A subexponential bound for linear programming. In: Proceedings of the 8th Annual Symposium on Computational Geometry, pp. 1–8 (1992)

  47. 47.

    Megiddo, N.: Linear programming in linear time when the dimension is fixed. J. Assoc. Comput. Mach. 31(1), 114–127 (1984)

  48. 48.

    Megiddo, N.: On the complexity of linear programming. In: Bewley, T. (ed.) Advances in Economic Theory: Fifth World Congress, pp. 225–268. Cambridge University Press, Cambridge (1987)

  49. 49.

    Naddef, D.: The Hirsch conjecture is true for (0,1)-polytopes. Math. Program. 45, 109–110 (1989)

  50. 50.

    Oda, T.: Convex Bodies and Algebraic Geometry. Springer, Berlin (1988)

  51. 51.

    Orlin, J.B.: A polynomial time primal network simplex algorithm for minimum cost flows. Math. Program. 78, 109–129 (1997)

  52. 52.

    Renegar, J.: A Mathematical View of Interior-Point Methods in Convex Optimization. SIAM, Philadelphia (2001)

  53. 53.

    Smale, S.: On the average number of steps of the simplex method of linear programming. Math. Program. 27, 241–262 (1983)

  54. 54.

    Smale, S.: Mathematical problems for the next century. In: Mathematics: Frontiers and Perspectives, pp. 271–294. American Mathematics Society, Providence (2000)

  55. 55.

    Spielman, D.A., Teng, S.: Smoothed analysis of algorithms: why the simplex algorithm usually takes polynomial time. J. ACM 51(3), 385–463 (2004)

  56. 56.

    Todd, M.J.: The monotonic bounded Hirsch conjecture is false for dimension at least 4. Math. Oper. Res. 5(4), 599–601 (1980)

  57. 57.

    Vershynin, R.: Beyond Hirsch conjecture: walks on random polytopes and smoothed complexity of the simplex method. In: IEEE Symposium on Foundations of Computer Science. NN, vol. 47, pp. 133–142. IEEE, New York (2006)

  58. 58.

    Walkup, D.W.: The Hirsch conjecture fails for triangulated 27-spheres. Math. Oper. Res. 3, 224–230 (1978)

  59. 59.

    Ziegler, G.M.: Lectures on Polytopes. Graduate Texts in Mathematics, vol. 152. Springer, Berlin (1995)

  60. 60.

    Ziegler, G.M.: Face numbers of 4-polytopes and 3-spheres. In: Proceedings of the International Congress of Mathematicians, vol. III, Beijing, 2002, pp. 625–634. Higher Ed. Press, Beijing (2002)

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Correspondence to Francisco Santos.

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E.D. Kim was supported in part by the Centre de Recerca Matemàtica, NSF grant DMS-0608785 and NSF VIGRE grants DMS-0135345 and DMS-0636297. F. Santos was supported in part by the Spanish Ministry of Science through grant MTM2008-04699-C03-02.

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Kim, E.D., Santos, F. An Update on the Hirsch Conjecture. Jahresber. Dtsch. Math. Ver. 112, 73–98 (2010). https://doi.org/10.1365/s13291-010-0001-8

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  • Graph diameter
  • Hirsch conjecture
  • Linear programming
  • Polytopes

Mathematics Subject Classification (2000)

  • 05C12
  • 52B05
  • 90C08