12 research outputs found
When is 0.999... equal to 1?
A doubly infinite sum, numerically evaluated at between 0.999 and 1.001,
turns out to have a nice value
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
Bit flipping and time to recover
We call `bits' a sequence of devices indexed by positive integers, where
every device can be in two states: (idle) and (active). Start from the
`ground state' of the system when all bits are in -state. In our first
Binary Flipping (BF) model, the evolution of the system is the following: at
each time step choose one bit from a given distribution on the
integers independently of anything else, then flip the state of this bit to the
opposite. In our second Damaged Bits (DB) model a `damaged' state is added:
each selected idling bit changes to active, but selecting an active bit changes
its state to damaged in which it then stays forever.
In both models we analyse the recurrence of the system's ground state when no
bits are active. We present sufficient conditions for both BF and DB models to
show recurrent or transient behaviour, depending on the properties of
. We provide a bound for fractional moments of the return time to
the ground state for the BF model, and prove a Central Limit Theorem for the
number of active bits for both models
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
Pivoting in Linear Complementarity: TwoPolynomial-Time Cases
We study the behavior of simple principal pivoting methods for the P-matrix linear complementarity problem (P-LCP). We solve an open problem of Morris by showing that Murty's least-index pivot rule (under any fixed index order) leads to a quadratic number of iterations on Morris's highly cyclic P-LCP examples. We then show that on K-matrix LCP instances, all pivot rules require only a linear number of iterations. As the main tool, we employ unique-sink orientations of cubes, a useful combinatorial abstraction of the P-LC