An Update on the Hirsch Conjecture
By Edward D. Kim, Francisco Santos. 2009
59 citations recorded by the source corpus.
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 n−d. 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 n−d 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 n−d 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.
Subfields: Applied Mathematics, Computer Graphics and Computer-Aided Design, Discrete Mathematics and Combinatorics
View Paper: https://www.semanticscholar.org/paper/2d5687420af57827df89d37279a74b4662888558
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In Applied Mathematics
In Computer Graphics and Computer-Aided Design
In Discrete Mathematics and Combinatorics
Citations in the corpus
- Nonlinear Discrete Optimization (2010, background): “Dantzig in 1957, and open to date, is whether the diameter of the graph of every polytope is polynomially bounded in its dimension and number of facets; see [61] for an up-to-date survey.”