130,085 research outputs found
Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut
We investigate the notion of stability proposed by Bilu and Linial. We obtain
an exact polynomial-time algorithm for -stable Max Cut instances with
for some absolute constant . Our
algorithm is robust: it never returns an incorrect answer; if the instance is
-stable, it finds the maximum cut, otherwise, it either finds the
maximum cut or certifies that the instance is not -stable. We prove
that there is no robust polynomial-time algorithm for -stable instances
of Max Cut when , where is the best
approximation factor for Sparsest Cut with non-uniform demands.
Our algorithm is based on semidefinite programming. We show that the standard
SDP relaxation for Max Cut (with triangle inequalities) is integral
if , where
is the least distortion with which every point metric space of negative
type embeds into . On the negative side, we show that the SDP
relaxation is not integral when .
Moreover, there is no tractable convex relaxation for -stable instances
of Max Cut when . That suggests that solving
-stable instances with might be difficult or
impossible.
Our results significantly improve previously known results. The best
previously known algorithm for -stable instances of Max Cut required
that (for some ) [Bilu, Daniely, Linial, and
Saks]. No hardness results were known for the problem. Additionally, we present
an algorithm for 4-stable instances of Minimum Multiway Cut. We also study a
relaxed notion of weak stability.Comment: 24 page
Robust randomized matchings
The following game is played on a weighted graph: Alice selects a matching
and Bob selects a number . Alice's payoff is the ratio of the weight of
the heaviest edges of to the maximum weight of a matching of size at
most . If guarantees a payoff of at least then it is called
-robust. In 2002, Hassin and Rubinstein gave an algorithm that returns
a -robust matching, which is best possible.
We show that Alice can improve her payoff to by playing a
randomized strategy. This result extends to a very general class of
independence systems that includes matroid intersection, b-matchings, and
strong 2-exchange systems. It also implies an improved approximation factor for
a stochastic optimization variant known as the maximum priority matching
problem and translates to an asymptotic robustness guarantee for deterministic
matchings, in which Bob can only select numbers larger than a given constant.
Moreover, we give a new LP-based proof of Hassin and Rubinstein's bound
Incrementally Learned Mixture Models for GNSS Localization
GNSS localization is an important part of today's autonomous systems,
although it suffers from non-Gaussian errors caused by non-line-of-sight
effects. Recent methods are able to mitigate these effects by including the
corresponding distributions in the sensor fusion algorithm. However, these
approaches require prior knowledge about the sensor's distribution, which is
often not available. We introduce a novel sensor fusion algorithm based on
variational Bayesian inference, that is able to approximate the true
distribution with a Gaussian mixture model and to learn its parametrization
online. The proposed Incremental Variational Mixture algorithm automatically
adapts the number of mixture components to the complexity of the measurement's
error distribution. We compare the proposed algorithm against current
state-of-the-art approaches using a collection of open access real world
datasets and demonstrate its superior localization accuracy.Comment: 8 pages, 5 figures, published in proceedings of IEEE Intelligent
Vehicles Symposium (IV) 201
On Robustness Analysis of a Dynamic Average Consensus Algorithm to Communication Delay
This paper studies the robustness of a dynamic average consensus algorithm to
communication delay over strongly connected and weight-balanced (SCWB)
digraphs. Under delay-free communication, the algorithm of interest achieves a
practical asymptotic tracking of the dynamic average of the time-varying
agents' reference signals. For this algorithm, in both its continuous-time and
discrete-time implementations, we characterize the admissible communication
delay range and study the effect of the delay on the rate of convergence and
the tracking error bound. Our study also includes establishing a relationship
between the admissible delay bound and the maximum degree of the SCWB digraphs.
We also show that for delays in the admissible bound, for static signals the
algorithms achieve perfect tracking. Moreover, when the interaction topology is
a connected undirected graph, we show that the discrete-time implementation is
guaranteed to tolerate at least one step delay. Simulations demonstrate our
results
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