457 research outputs found
An update on the Hirsch conjecture
The Hirsch conjecture was posed in 1957 in a letter 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.
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 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.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2
and put into the appendix arXiv:0912.423
Polyhedral graph abstractions and an approach to the Linear Hirsch Conjecture
We introduce a new combinatorial abstraction for the graphs of polyhedra. The
new abstraction is a flexible framework defined by combinatorial properties,
with each collection of properties taken providing a variant for studying the
diameters of polyhedral graphs. One particular variant has a diameter which
satisfies the best known upper bound on the diameters of polyhedra. Another
variant has superlinear asymptotic diameter, and together with some
combinatorial operations, gives a concrete approach for disproving the Linear
Hirsch Conjecture.Comment: 16 pages, 4 figure
Recent progress on the combinatorial diameter of polytopes and simplicial complexes
The Hirsch conjecture, posed in 1957, stated that the graph of a
-dimensional polytope or polyhedron with facets cannot have diameter
greater than . The conjecture itself has been disproved, but what we
know about the underlying question is quite scarce. Most notably, no polynomial
upper bound is known for the diameters that were conjectured to be linear. In
contrast, no polyhedron violating the conjecture by more than 25% is known.
This paper reviews several recent attempts and progress on the question. Some
work in the world of polyhedra or (more often) bounded polytopes, but some try
to shed light on the question by generalizing it to simplicial complexes. In
particular, we include here our recent and previously unpublished proof that
the maximum diameter of arbitrary simplicial complexes is in and
we summarize the main ideas in the polymath 3 project, a web-based collective
effort trying to prove an upper bound of type nd for the diameters of polyhedra
and of more general objects (including, e. g., simplicial manifolds).Comment: 34 pages. This paper supersedes one cited as "On the maximum diameter
of simplicial complexes and abstractions of them, in preparation
A counterexample to the Hirsch conjecture
The Hirsch Conjecture (1957) stated that the graph of a -dimensional
polytope with facets cannot have (combinatorial) diameter greater than
. That is, that any two vertices of the polytope can be connected by a
path of at most edges.
This paper presents the first counterexample to the conjecture. Our polytope
has dimension 43 and 86 facets. It is obtained from a 5-dimensional polytope
with 48 facets which violates a certain generalization of the -step
conjecture of Klee and Walkup.Comment: 28 pages, 10 Figures: Changes from v2: Minor edits suggested by
referees. This version has been accepted in the Annals of Mathematic
Not all simplicial polytopes are weakly vertex-decomposable
In 1980 Provan and Billera defined the notion of weak -decomposability for
pure simplicial complexes. They showed the diameter of a weakly
-decomposable simplicial complex is bounded above by a polynomial
function of the number of -faces in and its dimension. For weakly
0-decomposable complexes, this bound is linear in the number of vertices and
the dimension. In this paper we exhibit the first examples of non-weakly
0-decomposable simplicial polytopes
Graphs of Transportation Polytopes
This paper discusses properties of the graphs of 2-way and 3-way
transportation polytopes, in particular, their possible numbers of vertices and
their diameters. Our main results include a quadratic bound on the diameter of
axial 3-way transportation polytopes and a catalogue of non-degenerate
transportation polytopes of small sizes. The catalogue disproves five
conjectures about these polyhedra stated in the monograph by Yemelichev et al.
(1984). It also allowed us to discover some new results. For example, we prove
that the number of vertices of an transportation polytope is a
multiple of the greatest common divisor of and .Comment: 29 pages, 7 figures. Final version. Improvements to the exposition of
several lemmas and the upper bound in Theorem 1.1 is improved by a factor of
tw
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