29 research outputs found
Partial resampling to approximate covering integer programs
We consider column-sparse covering integer programs, a generalization of set
cover, which have a long line of research of (randomized) approximation
algorithms. We develop a new rounding scheme based on the Partial Resampling
variant of the Lov\'{a}sz Local Lemma developed by Harris & Srinivasan (2019).
This achieves an approximation ratio of , where is the minimum covering
constraint and is the maximum -norm of any column of the
covering matrix (whose entries are scaled to lie in ). When there are
additional constraints on the variable sizes, we show an approximation ratio of
(where is the maximum number
of non-zero entries in any column of the covering matrix). These results
improve asymptotically, in several different ways, over results of Srinivasan
(2006) and Kolliopoulos & Young (2005).
We show nearly-matching inapproximability and integrality-gap lower bounds.
We also show that the rounding process leads to negative correlation among the
variables, which allows us to handle multi-criteria programs
Cut Sparsification of the Clique Beyond the Ramanujan Bound: A Separation of Cut Versus Spectral Sparsification
We prove that a random -regular graph, with high probability, is a cut
sparsifier of the clique with approximation error at most , where
and denotes an error term that depends on and and goes to
zero if we first take the limit and then the limit .
This is established by analyzing linear-size cuts using techniques of
Jagannath and Sen derived from ideas in statistical physics, and analyzing
small cuts via martingale inequalities.
We also prove new lower bounds on spectral sparsification of the clique. If
is a spectral sparsifier of the clique and has average degree , we
prove that the approximation error is at least the "Ramanujan bound''
, which is met by -regular Ramanujan graphs,
provided that either the weighted adjacency matrix of is a (multiple of) a
doubly stochastic matrix, or that satisfies a certain high "odd
pseudo-girth" property. The first case can be seen as an "Alon-Boppana theorem
for symmetric doubly stochastic matrices," showing that a symmetric doubly
stochastic matrix with non-zero entries has a non-trivial eigenvalue of
magnitude at least ; the second case generalizes a
lower bound of Srivastava and Trevisan, which requires a large girth
assumption.
Together, these results imply a separation between spectral sparsification
and cut sparsification. If is a random -regular graph on
vertices, we show that, with high probability, admits a (weighted subgraph)
cut sparsifier of average degree and approximation error at most
, while every (weighted subgraph) spectral
sparsifier of having average degree has approximation error at least
.Comment: To appear in SODA 202
Efficient Flow-based Approximation Algorithms for Submodular Hypergraph Partitioning via a Generalized Cut-Matching Game
In the past 20 years, increasing complexity in real world data has lead to
the study of higher-order data models based on partitioning hypergraphs.
However, hypergraph partitioning admits multiple formulations as hyperedges can
be cut in multiple ways. Building upon a class of hypergraph partitioning
problems introduced by Li & Milenkovic, we study the problem of minimizing
ratio-cut objectives over hypergraphs given by a new class of cut functions,
monotone submodular cut functions (mscf's), which captures hypergraph expansion
and conductance as special cases.
We first define the ratio-cut improvement problem, a family of local
relaxations of the minimum ratio-cut problem. This problem is a natural
extension of the Andersen & Lang cut improvement problem to the hypergraph
setting. We demonstrate the existence of efficient algorithms for approximately
solving this problem. These algorithms run in almost-linear time for the case
of hypergraph expansion, and when the hypergraph rank is at most .
Next, we provide an efficient -approximation algorithm for finding
the minimum ratio-cut of . We generalize the cut-matching game framework of
Khandekar et. al. to allow for the cut player to play unbalanced cuts, and
matching player to route approximate single-commodity flows. Using this
framework, we bootstrap our algorithms for the ratio-cut improvement problem to
obtain approximation algorithms for minimum ratio-cut problem for all mscf's.
This also yields the first almost-linear time -approximation
algorithms for hypergraph expansion, and constant hypergraph rank.
Finally, we extend a result of Louis & Makarychev to a broader set of
objective functions by giving a polynomial time -approximation algorithm for the minimum ratio-cut problem based on
rounding -metric embeddings.Comment: Comments and feedback welcom
Hypergraph Diffusions and Resolvents for Norm-Based Hypergraph Laplacians
The development of simple and fast hypergraph spectral methods has been
hindered by the lack of numerical algorithms for simulating heat diffusions and
computing fundamental objects, such as Personalized PageRank vectors, over
hypergraphs. In this paper, we overcome this challenge by designing two novel
algorithmic primitives. The first is a simple, easy-to-compute discrete-time
heat diffusion that enjoys the same favorable properties as the discrete-time
heat diffusion over graphs. This diffusion can be directly applied to speed up
existing hypergraph partitioning algorithms.
Our second contribution is the novel application of mirror descent to compute
resolvents of non-differentiable squared norms, which we believe to be of
independent interest beyond hypergraph problems. Based on this new primitive,
we derive the first nearly-linear-time algorithm that simulates the
discrete-time heat diffusion to approximately compute resolvents of the
hypergraph Laplacian operator, which include Personalized PageRank vectors and
solutions to the hypergraph analogue of Laplacian systems. Our algorithm runs
in time that is linear in the size of the hypergraph and inversely proportional
to the hypergraph spectral gap , matching the complexity of
analogous diffusion-based algorithms for the graph version of the problem
Neutralino Dark Matter beyond CMSSM Universality
We study the effect of departures from SUSY GUT universality on the
neutralino relic density and both its direct detection and indirect detection,
especially by neutrino telescopes. We find that the most interesting models are
those with a value of lower than the universal case.Comment: 20 pages, 12 figures, JHEP format. Figures improved for B&W,
references added, typos and english correcte
Neutrino Decays over Cosmological Distances and the Implications for Neutrino Telescopes
We discuss decays of ultra-relativistic neutrinos over cosmological distances
by solving the decay equation in terms of its redshift dependence. We
demonstrate that there are significant conceptual differences compared to more
simplified treatments of neutrino decay. For instance, the maximum distance the
neutrinos have traveled is limited by the Hubble length, which means that the
common belief that longer neutrino lifetimes can be probed by longer distances
does not apply. As a consequence, the neutrino lifetime limit from supernova
1987A cannot be exceeded by high-energy astrophysical neutrinos. We discuss the
implications for neutrino spectra and flavor ratios from gamma-ray bursts as
one example of extragalactic sources, using up-to-date neutrino flux
predictions. If the observation of SN 1987A implies that \nu_1 is stable and
the other mass eigenstates decay with rates much smaller than their current
bounds, the muon track rate can be substantially suppressed compared to the
cascade rate in the region IceCube is most sensitive to. In this scenario, no
gamma-ray burst neutrinos may be found using muon tracks even with the full
scale experiment, whereas reliable information on high-energy astrophysical
sources can only be obtained from cascade measurements. As another consequence,
the recently observed two cascade event candidates at PeV energies will not be
accompanied by corresponding muon tracks.Comment: 20 pages, 6 figures, 1 table. Matches published versio
A First Search for coincident Gravitational Waves and High Energy Neutrinos using LIGO, Virgo and ANTARES data from 2007
We present the results of the first search for gravitational wave bursts
associated with high energy neutrinos. Together, these messengers could reveal
new, hidden sources that are not observed by conventional photon astronomy,
particularly at high energy. Our search uses neutrinos detected by the
underwater neutrino telescope ANTARES in its 5 line configuration during the
period January - September 2007, which coincided with the fifth and first
science runs of LIGO and Virgo, respectively. The LIGO-Virgo data were analysed
for candidate gravitational-wave signals coincident in time and direction with
the neutrino events. No significant coincident events were observed. We place
limits on the density of joint high energy neutrino - gravitational wave
emission events in the local universe, and compare them with densities of
merger and core-collapse events.Comment: 19 pages, 8 figures, science summary page at
http://www.ligo.org/science/Publication-S5LV_ANTARES/index.php. Public access
area to figures, tables at
https://dcc.ligo.org/cgi-bin/DocDB/ShowDocument?docid=p120000
Dark Matter Signals from Cascade Annihilations
A leading interpretation of the electron/positron excesses seen by PAMELA and
ATIC is dark matter annihilation in the galactic halo. Depending on the
annihilation channel, the electron/positron signal could be accompanied by a
galactic gamma ray or neutrino flux, and the non-detection of such fluxes
constrains the couplings and halo properties of dark matter. In this paper, we
study the interplay of electron data with gamma ray and neutrino constraints in
the context of cascade annihilation models, where dark matter annihilates into
light degrees of freedom which in turn decay into leptons in one or more steps.
Electron and muon cascades give a reasonable fit to the PAMELA and ATIC data.
Compared to direct annihilation, cascade annihilations can soften gamma ray
constraints from final state radiation by an order of magnitude. However, if
dark matter annihilates primarily into muons, the neutrino constraints are
robust regardless of the number of cascade decay steps. We also examine the
electron data and gamma ray/neutrino constraints on the recently proposed
"axion portal" scenario.Comment: 36 pages, 11 figures, 7 tables; references adde
Multi-messenger observations of a binary neutron star merger
On 2017 August 17 a binary neutron star coalescence candidate (later designated GW170817) with merger time 12:41:04 UTC was observed through gravitational waves by the Advanced LIGO and Advanced Virgo detectors. The Fermi Gamma-ray Burst Monitor independently detected a gamma-ray burst (GRB 170817A) with a time delay of ~1.7 s with respect to the merger time. From the gravitational-wave signal, the source was initially localized to a sky region of 31 deg2 at a luminosity distance of 40+8-8 Mpc and with component masses consistent with neutron stars. The component masses were later measured to be in the range 0.86 to 2.26 Mo. An extensive observing campaign was launched across the electromagnetic spectrum leading to the discovery of a bright optical transient (SSS17a, now with the IAU identification of AT 2017gfo) in NGC 4993 (at ~40 Mpc) less than 11 hours after the merger by the One- Meter, Two Hemisphere (1M2H) team using the 1 m Swope Telescope. The optical transient was independently detected by multiple teams within an hour. Subsequent observations targeted the object and its environment. Early ultraviolet observations revealed a blue transient that faded within 48 hours. Optical and infrared observations showed a redward evolution over ~10 days. Following early non-detections, X-ray and radio emission were discovered at the transient’s position ~9 and ~16 days, respectively, after the merger. Both the X-ray and radio emission likely arise from a physical process that is distinct from the one that generates the UV/optical/near-infrared emission. No ultra-high-energy gamma-rays and no neutrino candidates consistent with the source were found in follow-up searches. These observations support the hypothesis that GW170817 was produced by the merger of two neutron stars in NGC4993 followed by a short gamma-ray burst (GRB 170817A) and a kilonova/macronova powered by the radioactive decay of r-process nuclei synthesized in the ejecta
Astrophysical Origins of Ultrahigh Energy Cosmic Rays
In the first part of this review we discuss the basic observational features
at the end of the cosmic ray energy spectrum. We also present there the main
characteristics of each of the experiments involved in the detection of these
particles. We then briefly discuss the status of the chemical composition and
the distribution of arrival directions of cosmic rays. After that, we examine
the energy losses during propagation, introducing the Greisen-Zaptsepin-Kuzmin
(GZK) cutoff, and discuss the level of confidence with which each experiment
have detected particles beyond the GZK energy limit. In the second part of the
review, we discuss astrophysical environments able to accelerate particles up
to such high energies, including active galactic nuclei, large scale galactic
wind termination shocks, relativistic jets and hot-spots of Fanaroff-Riley
radiogalaxies, pulsars, magnetars, quasar remnants, starbursts, colliding
galaxies, and gamma ray burst fireballs. In the third part of the review we
provide a brief summary of scenarios which try to explain the super-GZK events
with the help of new physics beyond the standard model. In the last section, we
give an overview on neutrino telescopes and existing limits on the energy
spectrum and discuss some of the prospects for a new (multi-particle)
astronomy. Finally, we outline how extraterrestrial neutrino fluxes can be used
to probe new physics beyond the electroweak scale.Comment: Higher resolution version of Fig. 7 is available at
http://www.angelfire.com/id/dtorres/down3.html. Solicited review article
prepared for Reports on Progress in Physics, final versio