49 research outputs found
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particular, we include here our recent and previously unpublished proof that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 unified approach to complementarity in optimization
AbstractAn underlying general structure of complementary pivot theory is presented with applications to various problems in optimization theory. The applications include linear complementarity, fixed point theory, unconstrained and constrained convex optimization without derivatives, nonlinear complementarity, and saddle point problems
Geometric Combinatorics of Transportation Polytopes and the Behavior of the Simplex Method
This dissertation investigates the geometric combinatorics of convex
polytopes and connections to the behavior of the simplex method for linear
programming. We focus our attention on transportation polytopes, which are sets
of all tables of non-negative real numbers satisfying certain summation
conditions. Transportation problems are, in many ways, the simplest kind of
linear programs and thus have a rich combinatorial structure. First, we give
new results on the diameters of certain classes of transportation polytopes and
their relation to the Hirsch Conjecture, which asserts that the diameter of
every -dimensional convex polytope with facets is bounded above by
. In particular, we prove a new quadratic upper bound on the diameter of
-way axial transportation polytopes defined by -marginals. We also show
that the Hirsch Conjecture holds for classical transportation
polytopes, but that there are infinitely-many Hirsch-sharp classical
transportation polytopes. Second, we present new results on subpolytopes of
transportation polytopes. We investigate, for example, a non-regular
triangulation of a subpolytope of the fourth Birkhoff polytope . This
implies the existence of non-regular triangulations of all Birkhoff polytopes
for . We also study certain classes of network flow polytopes
and prove new linear upper bounds for their diameters.Comment: PhD thesis submitted June 2010 to the University of California,
Davis. 183 pages, 49 figure
A Combinatorial Certifying Algorithm for Linear Programming Problems with Gainfree Leontief Substitution Systems
Linear programming (LP) problems with gainfree Leontief substitution systems
have been intensively studied in economics and operations research, and include
the feasibility problem of a class of Horn systems, which arises in, e.g.,
polyhedral combinatorics and logic. This subclass of LP problems admits a
strongly polynomial time algorithm, where devising such an algorithm for
general LP problems is one of the major theoretical open questions in
mathematical optimization and computer science. Recently, much attention has
been paid to devising certifying algorithms in software engineering, since
those algorithms enable one to confirm the correctness of outputs of programs
with simple computations. In this paper, we provide the first combinatorial
(and strongly polynomial time) certifying algorithm for LP problems with
gainfree Leontief substitution systems. As a by-product, we answer
affirmatively an open question whether the feasibility problem of the class of
Horn systems admits a combinatorial certifying algorithm