522 research outputs found
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
A quasi-polynomial bound for the diameter of graphs of polyhedra
The diameter of the graph of a -dimensional polyhedron with facets is
at most Comment: 2 page
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particular, we include here our recent and previously unpublished proof that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 Quantitative Steinitz Theorem for Plane Triangulations
We give a new proof of Steinitz's classical theorem in the case of plane
triangulations, which allows us to obtain a new general bound on the grid size
of the simplicial polytope realizing a given triangulation, subexponential in a
number of special cases.
Formally, we prove that every plane triangulation with vertices can
be embedded in in such a way that it is the vertical projection
of a convex polyhedral surface. We show that the vertices of this surface may
be placed in a integer grid, where and denotes the shedding diameter of , a
quantity defined in the paper.Comment: 25 pages, 6 postscript figure
Witness (Delaunay) Graphs
Proximity graphs are used in several areas in which a neighborliness
relationship for input data sets is a useful tool in their analysis, and have
also received substantial attention from the graph drawing community, as they
are a natural way of implicitly representing graphs. However, as a tool for
graph representation, proximity graphs have some limitations that may be
overcome with suitable generalizations. We introduce a generalization, witness
graphs, that encompasses both the goal of more power and flexibility for graph
drawing issues and a wider spectrum for neighborhood analysis. We study in
detail two concrete examples, both related to Delaunay graphs, and consider as
well some problems on stabbing geometric objects and point set discrimination,
that can be naturally described in terms of witness graphs.Comment: 27 pages. JCCGG 200
On the Length of Monotone Paths in Polyhedra
Motivated by the problem of bounding the number of iterations of the Simplex
algorithm we investigate the possible lengths of monotone paths followed by the
Simplex method inside the oriented graphs of polyhedra (oriented by the
objective function). We consider both the shortest and the longest monotone
paths and estimate the monotone diameter and height of polyhedra. Our analysis
applies to transportation polytopes, matroid polytopes, matching polytopes,
shortest-path polytopes, and the TSP, among others. We begin by showing that
combinatorial cubes have monotone and Bland pivot height bounded by their
dimension and that in fact all monotone paths of zonotopes are no larger than
the number of edge directions of the zonotope. We later use this to show that
several polytopes have polynomial-size pivot height, for all pivot rules. In
contrast, we show that many well-known combinatorial polytopes have
exponentially-long monotone paths. Surprisingly, for some famous pivot rules,
e.g., greatest improvement and steepest edge, these same polytopes have
polynomial-size simplex paths.Comment: 24 pages, 8 figure
Diameter of Polyhedra: Limits of Abstraction
We investigate the diameter of a natural abstraction of the
-skeleton of polyhedra. Even if this abstraction is more general than
other abstractions previously studied in the literature,
known upper bounds on the diameter of polyhedra continue to hold
here. On the other hand, we show that this abstraction has its
limits by providing an almost quadratic lower bound
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