273 research outputs found
Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms
We present a technical survey on the state of the art approaches in data reduction and the coreset framework. These include geometric decompositions, gradient methods, random sampling, sketching and random projections. We further outline their importance for the design of streaming algorithms and give a brief overview on lower bounding techniques
FPT Approximation for Constrained Metric k-Median/Means
The Metric -median problem over a metric space is
defined as follows: given a set of facility locations
and a set of clients, open a set of
facilities such that the total service cost, defined as , is minimised. The metric -means
problem is defined similarly using squared distances. In many applications
there are additional constraints that any solution needs to satisfy. This gives
rise to different constrained versions of the problem such as -gather,
fault-tolerant, outlier -means/-median problem. Surprisingly, for many of
these constrained problems, no constant-approximation algorithm is known. We
give FPT algorithms with constant approximation guarantee for a range of
constrained -median/means problems. For some of the constrained problems,
ours is the first constant factor approximation algorithm whereas for others,
we improve or match the approximation guarantee of previous works. We work
within the unified framework of Ding and Xu that allows us to simultaneously
obtain algorithms for a range of constrained problems. In particular, we obtain
a -approximation and -approximation for the
constrained versions of the -median and -means problem respectively in
FPT time. In many practical settings of the -median/means problem, one is
allowed to open a facility at any client location, i.e., . For
this special case, our algorithm gives a -approximation and
-approximation for the constrained versions of -median and
-means problem respectively in FPT time. Since our algorithm is based on
simple sampling technique, it can also be converted to a constant-pass
log-space streaming algorithm
Observability of Dark Matter Substructure with Pulsar Timing Correlations
Dark matter substructure on small scales is currently weakly constrained, and
its study may shed light on the nature of the dark matter. In this work we
study the gravitational effects of dark matter substructure on measured pulsar
phases in pulsar timing arrays (PTAs). Due to the stability of pulse phases
observed over several years, dark matter substructure around the Earth-pulsar
system can imprint discernible signatures in gravitational Doppler and Shapiro
delays. We compute pulsar phase correlations induced by general dark matter
substructure, and project constraints for a few models such as monochromatic
primordial black holes (PBHs), and Cold Dark Matter (CDM)-like NFW subhalos.
This work extends our previous analysis, which focused on static or single
transiting events, to a stochastic analysis of multiple transiting events. We
find that stochastic correlations, in a PTA similar to the Square Kilometer
Array (SKA), are uniquely powerful to constrain subhalos as light as , with concentrations as low as that predicted by standard
CDM.Comment: 45 pages, 12 figure
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