231 research outputs found
Finding Short Paths on Polytopes by the Shadow Vertex Algorithm
We show that the shadow vertex algorithm can be used to compute a short path
between a given pair of vertices of a polytope P = {x : Ax \leq b} along the
edges of P, where A \in R^{m \times n} is a real-valued matrix. Both, the
length of the path and the running time of the algorithm, are polynomial in m,
n, and a parameter 1/delta that is a measure for the flatness of the vertices
of P. For integer matrices A \in Z^{m \times n} we show a connection between
delta and the largest absolute value Delta of any sub-determinant of A,
yielding a bound of O(Delta^4 m n^4) for the length of the computed path. This
bound is expressed in the same parameter Delta as the recent non-constructive
bound of O(Delta^2 n^4 \log (n Delta)) by Bonifas et al.
For the special case of totally unimodular matrices, the length of the
computed path simplifies to O(m n^4), which significantly improves the
previously best known constructive bound of O(m^{16} n^3 \log^3(mn)) by Dyer
and Frieze
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
On the Shadow Simplex Method for Curved Polyhedra
We study the simplex method over polyhedra satisfying certain âdiscrete curvatureâ lower bounds,
which enforce that the boundary always meets vertices at sharp angles. Motivated by linear
programs with totally unimodular constraint matrices, recent results of Bonifas et al (SOCG
2012), Brunsch and Röglin (ICALP 2013), and Eisenbrand and Vempala (2014) have improved
our understanding of such polyhedra.
We develop a new type of dual analysis of the shadow simplex method which provides a clean
and powerful tool for improving all previously mentioned results. Our methods are inspired by
the recent work of Bonifas and the first named author [4], who analyzed a remarkably similar
process as part of an algorithm for the Closest Vector Problem with Preprocessing.
For our first result, we obtain a constructive diameter bound of O( n2 ln n ) for n-dimensional polyhedra with curvature parameter 2 [0, 1]. For the class of polyhedra arising from totally
unimodular constraint matrices, this implies a bound of O(n3 ln n). For linear optimization,
given an initial feasible vertex, we show that an optimal vertex can be found using an expected O( n3 ln n ) simplex pivots, each requiring O(mn) time to compute. An initial feasible solutioncan be found using O(mn3 ln n ) pivot steps
Recommended from our members
Combinatorial Optimization (hybrid meeting)
Combinatorial Optimization deals with optimization problems defined on combinatorial structures such as graphs and networks. Motivated by diverse practical problem setups, the topic has developed into a rich mathematical discipline with many connections to other fields of Mathematics (such as, e.g., Combinatorics, Convex Optimization and Geometry, and Real Algebraic Geometry). It also has strong ties to Theoretical Computer Science and Operations Research. A series of Oberwolfach Workshops have been crucial for establishing and developing the field. The workshop we report about was a particularly exciting event - due to the depth of results that were presented, the spectrum of developments that became apparent from the talks, the breadth of the connections to other mathematical fields that were explored, and last but not least because for many of the particiants it was the first opportunity to exchange ideas and to collaborate during an on-site workshop since almost two years
Upper and Lower Bounds on the Smoothed Complexity of the Simplex Method
The simplex method for linear programming is known to be highly efficient in
practice, and understanding its performance from a theoretical perspective is
an active research topic. The framework of smoothed analysis, first introduced
by Spielman and Teng (JACM '04) for this purpose, defines the smoothed
complexity of solving a linear program with variables and constraints
as the expected running time when Gaussian noise of variance is
added to the LP data. We prove that the smoothed complexity of the simplex
method is , improving the dependence on
compared to the previous bound of .
We accomplish this through a new analysis of the \emph{shadow bound}, key to
earlier analyses as well. Illustrating the power of our new method, we use our
method to prove a nearly tight upper bound on the smoothed complexity of
two-dimensional polygons.
We also establish the first non-trivial lower bound on the smoothed
complexity of the simplex method, proving that the \emph{shadow vertex simplex
method} requires at least pivot steps with high probability. A key
part of our analysis is a new variation on the extended formulation for the
regular -gon. We end with a numerical experiment that suggests this
analysis could be further improved.Comment: 41 pages, 5 figure
Shadows of Newton Polytopes
We define the shadow complexity of a polytope P as the maximum number of vertices in a linear projection of P to the plane. We describe connections to algebraic complexity and to parametrized optimization. We also provide several basic examples and constructions, and develop tools for bounding shadow complexity
- âŠ