An Update on the Hirsch Conjecture

Abstract

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.

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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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Keywords

  • Graph diameter
  • Hirsch conjecture
  • Linear programming
  • Polytopes

Mathematics Subject Classification (2000)

  • 05C12
  • 52B05
  • 90C08