7 research outputs found
Two New Bounds on the Random-Edge Simplex Algorithm
We prove that the Random-Edge simplex algorithm requires an expected number
of at most 13n/sqrt(d) pivot steps on any simple d-polytope with n vertices.
This is the first nontrivial upper bound for general polytopes. We also
describe a refined analysis that potentially yields much better bounds for
specific classes of polytopes. As one application, we show that for
combinatorial d-cubes, the trivial upper bound of 2^d on the performance of
Random-Edge can asymptotically be improved by any desired polynomial factor in
d.Comment: 10 page
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
On the Existence of Hamiltonian Paths for History Based Pivot Rules on Acyclic Unique Sink Orientations of Hypercubes
An acyclic USO on a hypercube is formed by directing its edges in such as way
that the digraph is acyclic and each face of the hypercube has a unique sink
and a unique source. A path to the global sink of an acyclic USO can be modeled
as pivoting in a unit hypercube of the same dimension with an abstract
objective function, and vice versa. In such a way, Zadeh's 'least entered rule'
and other history based pivot rules can be applied to the problem of finding
the global sink of an acyclic USO. In this paper we present some theoretical
and empirical results on the existence of acyclic USOs for which the various
history based pivot rules can be made to follow a Hamiltonian path. In
particular, we develop an algorithm that can enumerate all such paths up to
dimension 6 using efficient pruning techniques. We show that Zadeh's original
rule admits Hamiltonian paths up to dimension 9 at least, and prove that most
of the other rules do not for all dimensions greater than 5
Grid Orientations, (d,d + 2)-Polytopes, and Arrangements of Pseudolines
We investigate the combinatorial structure of linear programs on simple d-polytopes with d + 2 facets. These can be encoded by admissible grid orientations. Admissible grid orientations are also obtained through orientation properties of a planar point configuration or by the dual line arrangement. The point configuration and the polytope corresponding to the same grid are related through an extended Gale transform. The class of admissible grid orientations is shown to contain nonrealizable examples, i.e., there are admissible grid orientations which cannot be obtained from a polytope or a point configuration. It is shown, however, that every admissible grid orientation is induced by an arrangement of pseudolines. This later result is used to prove several nontrivial facts about admissible grid orientation
Unique Sink Orientations of Grids
We introduce unique sink orientations of grids as digraph models for many well-studied problems, including linear programming over products of simplices, generalized linear complementarity problems over P-matrices (PGLCP), and simple stochastic games. We investigate the combinatorial structure of such orientations and develop randomized algorithms for finding the sink. We show that the orientations arising from PGLCP satisfy the Holt-Klee condition known to hold for polytope digraphs, and we give the first expected linear-time algorithms for solving PGLCP with a fixed number of block