839 research outputs found
Violator Spaces: Structure and Algorithms
Sharir and Welzl introduced an abstract framework for optimization problems,
called LP-type problems or also generalized linear programming problems, which
proved useful in algorithm design. We define a new, and as we believe, simpler
and more natural framework: violator spaces, which constitute a proper
generalization of LP-type problems. We show that Clarkson's randomized
algorithms for low-dimensional linear programming work in the context of
violator spaces. For example, in this way we obtain the fastest known algorithm
for the P-matrix generalized linear complementarity problem with a constant
number of blocks. We also give two new characterizations of LP-type problems:
they are equivalent to acyclic violator spaces, as well as to concrete LP-type
problems (informally, the constraints in a concrete LP-type problem are subsets
of a linearly ordered ground set, and the value of a set of constraints is the
minimum of its intersection).Comment: 28 pages, 5 figures, extended abstract was presented at ESA 2006;
author spelling fixe
Counting Unique-Sink Orientations
Unique-sink orientations (USOs) are an abstract class of orientations of the
n-cube graph. We consider some classes of USOs that are of interest in
connection with the linear complementarity problem. We summarise old and show
new lower and upper bounds on the sizes of some such classes. Furthermore, we
provide a characterisation of K-matrices in terms of their corresponding USOs.Comment: 13 pages; v2: proof of main theorem expanded, plus various other
corrections. Now 16 pages; v3: minor correction
The Niceness of Unique Sink Orientations
Random Edge is the most natural randomized pivot rule for the simplex
algorithm. Considerable progress has been made recently towards fully
understanding its behavior. Back in 2001, Welzl introduced the concepts of
\emph{reachmaps} and \emph{niceness} of Unique Sink Orientations (USO), in an
effort to better understand the behavior of Random Edge. In this paper, we
initiate the systematic study of these concepts. We settle the questions that
were asked by Welzl about the niceness of (acyclic) USO. Niceness implies
natural upper bounds for Random Edge and we provide evidence that these are
tight or almost tight in many interesting cases. Moreover, we show that Random
Edge is polynomial on at least many (possibly cyclic) USO. As
a bonus, we describe a derandomization of Random Edge which achieves the same
asymptotic upper bounds with respect to niceness and discuss some algorithmic
properties of the reachmap.Comment: An extended abstract appears in the proceedings of Approx/Random 201
Finding the Sink Takes Some Time: An Almost Quadratic Lower Bound for Findingthe Sink of Unique Sink Oriented Cubes
We give a worst-case Ω(n 2/log n) lower bound on the number of vertex evaluations a deterministic algorithm needs to perform in order to find the (unique) sink of a unique sink oriented n-dimensional cube. We consider the problem in the vertex-oracle model, introduced in [18]. In this model one can access the orientation implicitly, in each vertex evaluation an oracle discloses the orientation of the edges incident to the queried vertex. An important feature of the model is that the access is indeed arbitrary, the algorithm does not have to proceed on a directed path in a simplex-like fashion, but could "jump around”. Our result is the first superlinear lower bound on the problem. The strategy we describe works even for acyclic orientations. We also give improved lower bounds for small values of n and fast algorithms in a couple of important special classes of orientations to demonstrate the difficulty of the lower bound proble
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
Reconstructing a Simple Polytope from its Graph
Blind and Mani (1987) proved that the entire combinatorial structure (the
vertex-facet incidences) of a simple convex polytope is determined by its
abstract graph. Their proof is not constructive. Kalai (1988) found a short,
elegant, and algorithmic proof of that result. However, his algorithm has
always exponential running time. We show that the problem to reconstruct the
vertex-facet incidences of a simple polytope P from its graph can be formulated
as a combinatorial optimization problem that is strongly dual to the problem of
finding an abstract objective function on P (i.e., a shelling order of the
facets of the dual polytope of P). Thereby, we derive polynomial certificates
for both the vertex-facet incidences as well as for the abstract objective
functions in terms of the graph of P. The paper is a variation on joint work
with Michael Joswig and Friederike Koerner (2001).Comment: 14 page
- …