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Welcome to Alpha-1 News from around the world. We find the news so you don't have to.
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New promotion, New Board Member and New website!
In: Charity News
We at Alpha-1 Awareness have some wonderful news that we would like to share with you. Internal Promotion Glyn Cockaday joined our charity Board 3 years ago as our Facebook Admin. Glyn not only excelled at this role, but started to help with many other internal projects and often tookContinue Reading
The sad passing of Dawn Haywood- Jones
On: 21 March 2019
It is with great sadness that we have to write about the sad passing of Dawn Haywood- Jones. Dawn along with 3 other people started Alpha-1 Awareness in 2008 in the hope that they could reach out to fellow sufferers and to educate the medical profession on this Rarely knownContinue Reading
Alpha-1 Awareness donates £10,000 to the research of a cure
On: 27 February 2019
In: Charity News, Research - Treatment & Trials
Alpha-1 Awareness is dedicated to contributing a minimum of 10% of its funds raised towards research to find a cure for this condition. In 2018, we went above and beyond this target, donating £25,000 to the patron of the Charity, Prof Lomas, Vice-Provost of Health at University College London (UCL).Continue Reading
Alpha-1 Awareness visits Mereo BioPharma
On: 20 December 2018
On Monday 17th December we went to see Mereo BioPharma at their office in London. Mereo BioPharma have recently started dosing in their Phase 2 clinical trial for MPH-966 (alvelestat). The study will recruit in America, Canada, Europe and the UK. The plan is that people who are PiZZContinue Reading
Alpha-1 Awareness Visit the people at UCL who are looking for a cure for Alpha-1
On: 28 July 2018
On the 26th July 2018, Alan Wooler went to see some of the team at UCL who are doing the vital research into finding a cure for Alpha-1. Alistair Jagger who is a PhD student in Prof Lomas' team was kind enough to show Alan around the labs and answerContinue Reading
We celebrate being the oldest Alpha-1 UK charity with our 10th Year Celebration Day party
On 8 July 2018, Alpha-1 Awareness hosted its 10 year Celebration Day, celebrating 10 years since the founding of the charity. World-renowned speakers were invited to share their knowledge on Alpha-1 Antitrypsin Deficiency, Professor David Lomas and Professor Nedim Hadzic, as well as those conquering the condition themselves, Karen Skålvoll.Continue Reading
New Children's club
On: 1 July 2018
EXCITING NEWS…. We have now launched the new Alphie Kids Club. The first club for children affected by Alpha-1 where they can come along and download colouring in pages, puzzles and become a member and receive our exclusive membership pack. So please click on the link and join your childrenContinue Reading
Our chairman Alan Wooler chosen as "Alpha-1 Leader in Action" in the latest Alpha-1 Global Newsletter
On: 8 June 2018
8th June 2018 Our charity is here to bring awareness of AATD and to support those affected. We are making a big difference and many people rely on our help. It is wonderful when other groups help to tell others who we are and what we do. In the latestContinue Reading
Alpha-1 Awareness UK changes charity name
3rd June 2018 Our charity was originally created to support people affected in the UK with AATD. It didn't take long before we started to help those from other countries and today we stretch as far as Australia and New Zealand. It is for this reason that we want toContinue Reading
Our charity donates £25,000 to the research of a cure
On: 26 January 2018
On the 20th January 2018 Alpha-1 Awareness UK went to London to see Professor David Lomas. We were there to make a £25,000 (yes, Twenty Five Thousand pounds) donation to go to the research of a cure for Alpha-1. We were shown around his different laboratories and inside many ofContinue Reading
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None of the articles that appear in this website have been written by us. We cannot confirm if what they state is accurate. Alpha-1 Awareness cannot be held responsible for any misleading information contained in any of the articles that we have posted on this site.
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In advancing VLSI technology, OFDM has received a great deal of attention and been adopted in many new generation wideband data communication systems. In this paper the analysis of several FFT algorithms such as radix-2, radix-4, split radix and mixed radix 4-2 were designed using VHDL. The results show that the proposed processor architecture can greatly save the area cost while keeping a high- speed processing speed, which may be attractive for many real-time systems. By using the proposed structure, the proposed algorithm makes an offer the simple bit reversal mechanism which is only supported by a fixed radix FFT algorithm.


A typical student of the NE Language Project is a professional or company employee who is looking to improve their language for their career or as an extra skill, ages of students can range from 18 to 80.
Some younger learners study with us and tutors always have the choice to accept or refuse work.
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Gottlieb '99 joins NBA's Cavaliers in historic hiring
University to require sophomores to stay on meal plan
University releases third annual DIAP progress report
Senior Orators: Ruth Miller '19, Patricia Rodarte '19 MD'23
Studies show ethnic pride linked to success in youth
Meta-analysis concludes positive feelings toward ethnicity improve academics, well-being
By Molly Schulson
Senior Staff Writer
Researchers combed through the abstracts of over 6,000 studies to analyze youth ethnic-racial affect — how positive a person feels about their own race or ethnicity — in a meta-analysis published Feb. 3 in the journal Child Development. They concluded that feelings of ethnic pride and happiness in minority youths positively affect their behavior and academic success.
Deborah Rivas-Drake, a former assistant professor of education and human development who is now an associate professor at the University of Michigan, and Adriana Umana-Taylor, professor at Arizona State University, formed the Ethnic and Racial Identity in the 21st Century working group, which authored the paper, "to incorporate diverse perspectives" on ethnic and racial identity, Rivas-Drake wrote in an email to The Herald.
In two meetings, held May 2012 and January 2013, the group discussed what they knew about racial-ethnic identity research in adolescents and identified gaps in the literature, said Tiffany Yip, associate professor of psychology at Fordham University and a member of the group. The meta-analysis sought to fill these gaps.
Synthesizing studies
In a meta-analysis, "you do a thorough review of all types of research on a particular topic and you analyze those studies," Yip said. "Instead of collecting data from people, your data points are other people's research papers."
The working group picked 46 studies eligible for inclusion in the meta-analysis, Rivas-Drake wrote.
Single studies sometimes restrict researchers from generalizing their findings so "it is important from time to time to synthesize the findings from all available studies," she wrote.
Marley Pierce '13, Rivas-Drake's research assistant for the project, double-checked that articles focused on youths, took place in the United States and discussed positive ethnic-racial affect. "My job was to go through articles a little more deeply to make sure they fit," she said.
Pride matters
"Across all of these studies, (researchers) found that having a positive sense of identity and feeling proud about your ethnic-racial group led to a host of positive measures," Yip said.
Results from the meta-analysis showed that positive ethnic-racial affect correlated with measures of high self-esteem, well-being and academic adjustment, and was negatively related to factors such as anxiety.
"We were able to synthesize results of studies across a very broad spectrum of indicators of youth development," Rivas-Drake wrote. "We see similar benefits across this broad spectrum."
While it may be obvious that a positive affect relates to higher self-esteem, it is less obvious that it negatively correlates with substance abuse or risky behaviors, Yip said. "I think the breadth of the outcomes is an important contribution," she added.
The majority of studies in the meta-analysis examined middle school and high school students because most psychological theories support the idea that ethnic-racial identity begins to take shape during adolescence, Yip said.
About half the studies included black youths, but Hispanic, American Indian and Asian American and Pacific Islander youths were also represented.
But according to the meta-analysis, findings were consistent across age, gender and ethnic-racial groups.
The meta-analysis "offers one of the most highly sophisticated analyses over age and developmental levels that exist in the field on ethnic and racial identity and the phenomenon of exclusionary practices," said Lewis Lipsitt, professor emeritus of cognitive, linguistic and psychological sciences, who was not involved in the study.
The findings will "really move the field forward," said Aerika Brittian, assistant professor of educational psychology at the University of Illinois at Chicago, who was also not involved in the study. In the future, more longitudinal research on different aspects of identity would be beneficial, she added.
The working group may consider adding a longitudinal component to future endeavors, Yip said, though such studies present logistical and financial challenges.
"I would also like to see more work done in different parts of the country. I think context is really important," Brittian said. For example, ethnic-racial identity for minorities in Los Angeles functions differently than it does in Chicago, she added.
Pierce said though some of the findings may seem obvious, the researchers' "statistically significant" correlations are important for the field.
As a result of this meta-analysis, there is a lot schools can do, such as providing the opportunity for students to learn about their cultural groups, Brittian said. "Educators can consider how to facilitate positive affect … in a classroom setting," she added.
In the future, Rivas-Drake wants to examine how positive ethnic-racial identity is supported in schools and local communities, she wrote.
Lipsitt said he hopes the meta-analysis will spark further discussion.
"As long as the talk continues, inspired by good science, I think we can't go wrong in developing a further and complete understanding on how it is growing up under conditions of group exclusion," he said.
Panel says challenges remain for those of multiracial identity
Experts square off on race representation
Study finds minorities receive subpar anxiety care
Study evaluates patient-doctor interactions
Honorary degree recipients 2019
Brain science blooms at Brown
Mark Liang: On writing
Open Curriculum at 50

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To celebrate International Women's Day 2019 we've collaborated with other female designers on two new versions of our Bite Back design, £5 from each sale of these sweatshirts will go to Women For Women International.
We have collaborated with the brilliant Emotional Waterfall Art to create these 'Bite back' sweatshirts featuring the Bristol based artist's hand painted fabric.
Each design is appliquéd by our talented seamstresses in Brighton UK using hand cut panels of metallics, satins and hand painted cotton. Our sweatshirts are an 80 / 20 cotton mix and are all WRAP certified fair-trade.
Please note that the pattern on each sweatshirt will look different as this fabric is hand painted, although the base colour and over all look will be the same as pictured.


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The Master of Science degree program in the Department of Biomedical Engineering (BME) at Columbia University offers world-class biomedical engineering education and opportunities for cutting edge research and innovation.
The mission of the MS program is to prepare talented students from diverse backgrounds to become innovative, socially responsible leaders in engineering, science, and medicine. Graduates from our MS program have landed top positions in the medical device industry and at engineering consulting firms, have founded their own medical technology companies, and have gone on to advanced graduate studies in engineering and professional studies in medicine. The MS degree program in the Department of BME at Columbia University offers a surprising mix of the intellectual atmosphere of an Ivy League institution and the sense of community of a small college, enriched by the diversity of New York City.
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The MS degree program in the Department of BME prepares students for research activities at the highest level. 90% of our MS students participate in cutting edge research during the course of their degree program. The Department of Biomedical Engineering also maintains a strong partnership with the Columbia University Medical Center, fostering interdisciplinary collaborations that add to the quality and strength of the program. Research areas of particular interest include biomechanics, cellular and tissue engineering, artificial organs, regenerative medicine, auditory biophysics, biosignals and biomedical imaging, point-of-care diagnostics, and neural engineering. The MS degree program is well equipped to prepare students for admission to highly competitive doctoral programs across the United States, as well as employment within industry fields of biotechnology and medical devices.
"This flexible program allowed me to sample many fields underneath the biomedical engineering umbrella while having the opportunity to conduct mentored research either for credit towards my degree or as a volunteer. The coursework I completed and research experience I received were what prepared me best for the life of research I have chosen to pursue and helped build the strong resume I needed to apply to top tier doctoral programs."
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MS '15, PhD candidate Columbia University
The MS degree program in the Department of BME provides students with unparalleled training and mentorship in medical technology innovation and entrepreneurship. Project-based courses in the design and commercialization of medical innovations offered by the Department of Biomedical Engineering, complemented by courses offered by other departments and schools, including the Columbia Business School and Columbia University Medical Center, provide students with a rich interdisciplinary educational experience. Graduates of our MS degree program are able to contribute to and drive medical innovation and support the promotion, development, and realization of the clinical potential of translational research.
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MS degree students in the Department of BME come from diverse training, including research and industry, and backgrounds, including sciences (such as physics, chemistry, and biology), mathematics, and several fields of engineering (such as mechanical, electrical, and chemical engineering). The program offers a flexible curriculum that allows students interested in transitioning into biomedical engineering the opportunity to individualize their experience and complete the necessary requirements to prepare for advanced study at the graduate-level. MS degree students work closely with their Faculty advisor to identify courses and research/project experiences that will allow for a seamless transition into the field of BME. The program also provides excellent preparation for students interested in transitioning into the health sciences and the study of medicine.
All applicants to our MS program should have a solid foundation in mathematics, including multivariate calculus, linear algebra, and differential equations and experience in computer programming (Matlab or Python).
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\section{Introduction}
\seclab{intro}

\paragraph{Motivation and examples.}

Consider the problem of \emphi{obnoxious facility location}
\cite{t-oflg-91, C-sofl-99}; that is, you have to place several
facilities, but these facilities are undesired (i.e.,
obnoxious). Facilities of this type include nuclear reactors, wind
farms, airports, power plants, factories, prisons, universities,
etc. Facilities can also be semi-desirable -- a customer might want to
have supermarkets close to their home, but they do not want to have
too many of them close by as they increase traffic, noise, etc. One
natural way to model this geometrically is to associate each obnoxious
facility with its region of undesirability. We also have customers
(modeled as points), and each customer has a threshold of how many
obnoxious facilities it is willing to accept covering it. Different
customers may have different thresholds, for example because more
affluent people have stronger political power and it is harder to
place obnoxious facilities near their homes.

Naturally, if you allow only a single region to cover each customer,
then this is a classical packing problem, and much work has been done
on packing disks/balls \cite{smcscg-nacps-07}. However, there are many
cases where allowing limited interaction between the packed regions is
allowed (after all, these facilities are required for modern existence).
As a concrete example of this type of problem, consider the placement
of radio stations/cellphone towers. While airports allow only very
limited levels of interference\footnote{See
   \si{\url{\si{http://tinyurl.com/7td67v3}}} for a story of an
   airport closing down because of radio interference.}, higher levels
of such interference is acceptable in residential neighborhoods.  However,
at a certain point there is going to be resistance to placing
more wireless towers in residential areas, as these towers are viewed as
causing cancer (this fear might be baseless, but it does not change the political reality of the difficulty of placing such towers). On the other
hand, there is little resistance to placing such towers along highways in sparsely populated areas.

In this paper, we are interested in the modeling of such problems and
in the computation of an efficient approximation to the optimal
solution of such problems.

\subsection*{Modeling.}
As hinted by the above, perhaps the most natural way to model this problem is
as a generalization of the well known independent set problem.

\paragraph{Independent set} %
is a fundamental discrete optimization problem.  Unfortunately, it is
not only computationally hard, but it is even hard to approximate to
within a factor of $n^{1-\eps}$, for any constant $\eps$
\cite{h-chaw-99} (under the assumption that
$\textsf{NP}\neq\textsf{P}$). Surprisingly, the problem is
considerably easier in some geometric settings. For example, there is
a \Term{P{T}AS}\xspace\footnote{Polynomial time approximation scheme.}
\cite{c-ptasp-03,ejs-ptasg-05} for the following problem: Given a set
of unit disks in the plane, find a maximum cardinality subset of the
disks whose interiors are disjoint.  Furthermore, a simple local
search algorithm yields the desired approximation: For any $\eps > 0$,
the local search algorithm that tries to swap subsets of size $O(1 /
\eps^2)$ yields a $(1 - \eps)$-approximation in $n^{O(1/\eps^2)}$ time
\cite{ch-aamis-09, ch-aamis-11}.

\paragraph{The discrete independent set problem.} %
In this paper, we consider packing problems in geometric settings that
are natural extensions of the geometric independent set problem
described above.  As a starting point, motivated by practical
applications, we consider the discrete version of the geometric
independent set problem in which, in addition to a set of weighted
regions, we are given a set of points, and the goal is to select a
maximum weight subset of the regions so that each point is contained
in at most one of the selected regions.  We refer to this problem as
the \emphi{discrete independent set} problem.  Chan and Har-Peled
\cite{ch-aamis-11} studied this discrete variant and proved that one
can get a good approximation if the union complexity of the regions is
small.

\parpic[r]{\includegraphics{figs/flower}}%

Note that the discrete independent set problem captures the continuous
version of the independent set problem, since we can place a point in
each face of the induced arrangement of the given regions.  In fact,
the discrete version is considerably harder (in some cases) than the
continuous variant.  The difficulty lies in that several regions
forming a valid solution to an instance of a discrete independent set
problem may contain a common point that is not part of the set of
points given as input; the figure on the right shows an example in
which the middle point, marked as a square, is covered twice by the
given valid solution.

\parpic[r]{\includegraphics{figs/k_t_t}}%

To illustrate the difference in difficulty, consider the case when the
input consists of a set $\SegSet$ of segments (in general position)
with their endpoints on a circle, such that every pair of segments
intersect.  Clearly, in the continuous version, the maximum
independent set of segments is a single segment.  However, in this
case, the discrete version captures the graph independent set
problem. More precisely, we can encode any instance of independent set
(i.e., a graph $G=(V,E)$) as an instance of this problem as
follows. Every vertex $v \in V$ is mapped to a segment $\seg_v$ of
$\SegSet$, and every edge $uv \in E$, is mapped to the point $\seg_u
\cap \seg_v$ (which is added to a set of points $\PntSet$). Clearly,
an independent set of segments of $\SegSet$ (in relation to the point
set $\PntSet$) corresponds to an independent set in $G$. That is, the
geometric discrete version is sometimes as hard as the graph
independent set problem.  For example, the figure on the right depicts
the resulting instance encoding independent set for $K_{3,3}$.

\paragraph{The packing problem.} %
In this paper, we are interested in the natural extension of the
discrete independent set problem to the case where every point has a
capacity and might be covered several times (but not exceeding its
capacity). The resulting problem has a flavor of a packing problem,
and is defined formally as follows.

\begin{problem} {\rm{(\PackRegions.)}} %
    Given a set $\ObjSet$ of regions and a set $\PntSet$ of points
    such that each region $r$ has a weight $\weight{r}$ and each point
    ${\mathsf{p}}$ has a capacity $\capacityX{{\mathsf{p}}}$, find a maximum weight
    subset $X$ of the regions such that, for each point ${\mathsf{p}}$, the
    number of regions in $X$ that contain ${\mathsf{p}}$ is at most its
    capacity $\capacityX{{\mathsf{p}}}$.

    \problab{pack:regions}
\end{problem}

We emphasize that different points might have different capacities,
which makes the problem considerably more challenging to solve than
the unit capacities case (i.e., the discrete independent set problem).
We also consider the following dual problem in which the points have
weights and the regions have capacities.

\begin{problem}{\rm{(\PackPoints.)}}
    Given a set $\ObjSet$ of regions and a set $\PntSet$ of points
    such that each region $r$ has a capacity $\capacityX{r}$ and each
    point ${\mathsf{p}}$ has a weight $\weight{{\mathsf{p}}}$, find a maximum weight
    subset $X$ of the points such that each region $r$ contains at
    most $\capacityX{r}$ points of $X$.

    \problab{pack:points}
\end{problem}

\noindent
\textbf{Hypergraph framework.} %
These two problems can be stated in a unified way in the language of
hypergraphs\footnote{A \emphi{hypergraph} $\hgraph$ is a pair
   $(\VSet,\HESet)$, where $\VSet$ is a set of vertices and $\HESet$
   is a collection of subsets of $\VSet$ which are called
   \emphi{hyperedges}.}. Given an instance of \PackRegions, we
construct a hypergraph as follows: Each weighted region is a vertex,
and all the regions containing a given point of capacity $k$ become a
hyperedge (consisting of these regions) of capacity $k$.  A similar
reduction works for \PackPoints, where the given weighted points are
the vertices, and each region of capacity $k$ becomes a hyperedge of
capacity $k$ consisting of all of the points contained in this
region. Therefore the previous two problems are special cases of the
following problem.

\begin{problem}{\rm{(\PackHGraph.)}}
    Given a hypergraph $\hgraph = (\VSet,\HESet)$ with a weight
    function $\weight{\cdot}$ on the vertices and a capacity function
    $\capacityX{\cdot}$ on the hyperedges, find a maximum weight
    subset $X\subseteq \VSet$, such that $\forall \hedge\in \HESet$ we
    have $|X\cap \hedge|\leq \capacityX{\hedge}$.
\end{problem}

We will be interested primarily in hypergraphs with certain hereditary
properties. A hypergraph property is \emph{hereditary} if the
sub-hypergraph induced by any subset of the vertices has the property;
an example of a hereditary property of hypergraphs is having bounded
\Term{VC}\xspace dimension. Roughly, we are interested in hypergraphs having the
\emphi{bounded growth property}: For any induced sub-hypergraph on $t$
vertices the number of its hyperedges that contain exactly $k$
vertices is near linear in $t$ and its dependency on $k$ is bounded by
$2^{O(k)}$, see \defref{bounded}. Such hypergraphs arise naturally
when considering points and ``nice'' regions in the plane.

\subsection*{Our results.}
\begin{compactitem}[$\bullet$]
    \item \xparagraph{Main result.} Our main result is an algorithm
    that provides a good approximation for \PackHGraph as a function
    of the growth of the hypergraph, see \thmref{p:hypergraph}.  Our
    result can be viewed as an extension of the work of Chan and
    Har-Peled \cite{ch-aamis-11} to these considerably more general
    and intricate settings.

    \item \xparagraph{Regions with low union complexity.}  In
    \secapndref{applications}, we apply our main result to regions that
    have low union complexity, and we get the following results:

    \begin{compactenum}[(A)]
        \item If the union complexity of $n$ regions is $O( n u(n))$
        then we get an $O\pth{ u(n)^{1/\nu} }$-approximation for
        \PackRegions, where $\nu$ is the minimum capacity of any point
        in the given instance. (That is, the problem becomes easier as
        the minimum capacity increases.)  For the case where all the
        capacities are one, this is the discrete independent set
        problem, and our algorithm specializes to the algorithm of
        Chan and Har-Peled \cite{ch-aamis-11}, which gives an
        $O(u(n))$-approximation.

        \item More specifically, we get a constant factor approximation
        for \PackRegions if the union complexity of the regions is
        linear. This holds for
        \begin{inparaenum}[(i)]
            \item fat-triangles of similar size,
            \item unit axis-parallel cubes in 3d, and
            \item pseudo-disks.
        \end{inparaenum}
        See \corref{pseudo}.

        \item Similarly, since the union complexity of fat triangles
        in the plane is $O(n \log^* n )$ \cite{eas-ibuft-11, abes-ibufop-11},
        we get an $O\pth{ \pth{\log^* n}^{1/\nu} }$ approximation for
        such instances of \PackRegions.
    \end{compactenum}

    \item \xparagraph{Bi-criteria approximation.} %
    Our main result also implies a bi-criteria approximation
    algorithm. That is, we can improve the quality of the solution, at
    the cost of potentially violating low capacity regions.  Formally,
    if the input instance $\hgraph = (\VSet, \HESet)$ of \PackHGraph
    has at most $\feq{k}{t} = 2^{O(k)} F(t)$ edges of size $k$ when
    restricted to any subset of $t$ vertices, then for any integer
    $\phi \geq 1$, our algorithm yields an $\pth{
       O\pth{(F(n)/n)^{1/\phi}}, \phi} $-approximation to the
    given instance $\hgraph$ of \PackHGraph.  Specifically, the value
    of the generated solution $\VSetA$ is at least $\Omega\pth{ \Opt/
       (F(n)/n)^{1/\phi}}$, where $\Opt$ is the value of the
    optimal solution, and for every hyperedge $\hedge \in \HESet$, we
    have $\cardin{\hedge \cap \VSetA} \leq \max \pth{ \phi,
       \capacityX{\hedge}}$.

    As an example, for any set of $n$ regions in the plane such that
    the boundaries of any pair of them intersects $O(1)$ times, the
    above implies that one can get an $\pth{O\pth{n^{1/\phi}}, \,
       \phi}$-approximation for \PackRegions.

    \item \xparagraph{Axis-parallel boxes.} The union complexity of
    axis-parallel rectangles can be as high as quadratic, and
    therefore we cannot immediately apply our main result to get a
    good approximation. Instead, we decompose the union of
    axis-parallel rectangles into regions of low union complexity, and
    this decomposition together with our main result gives us an
    $O(\log n)$ approximation for instances of \PackRegions in which
    the regions are axis-parallel rectangles in the plane%
    (see \lemref{p:rect:in:pnts}).

    A more involved analysis also applies to the three dimensional
    case, where we get an $O(\log^3n)$ approximation for \PackRegions
    for axis parallel boxes (see \lemref{p:boxes:in:pnts}).

    \item \xparagraph{Dual problem.}  We show in
    \secapndref{halfRayDisk} that, by standard lifting techniques, we
    can apply our result for \PackRegions, where the regions are
    disks, to the dual problem of \PackPointsInDisks.  However, for
    other regions, the dual problem \PackPoints seems to be more
    challenging.  Specifically, this is true for the case of
    axis-parallel rectangles.  For this case, we first provide a
    constant factor approximation for \emph{skyline} instances of the
    problem; a skyline instance is a set of rectangles that lie on the
    $x$-axis. Interestingly, if the set of rectangles are defined in
    relation to a set of points (and each rectangles contains only a
    few points), then one can define a near-linear (in the number of
    points) sized set of rectangles such that each original rectangle
    is the union of two new rectangles. Combining this with the
    skyline result and a sparsifying technique, we get an $\pth{ O(
       \log n), 2}$-approximation; that is, every rectangle $\mathsf{b}$
    contains at most $\max(2, \capacityX{\mathsf{b}})$ points of the
    solution constructed, and the total weight of the solution is
    $\Omega(\Opt/\log n)$%
    (see \thmref{pack:points:in:rects}). (Note that, by applying our
    general framework directly to this setting, we only get an $\pth{
       O\pth{n^{1/\phi}}, \phi}$-approximation, for any integer
    $\phi > 0$.)

    \item \xparagraph{Packing points into fat triangles.}  %
    We provide a polylog bi-criteria approximation for the problem of
    packing points into fat triangles. This requires proving that one
    can compute, for a given point set, a small number of canonical
    subsets, such that the point set covered by any fat-triangle (if
    the set is sufficiently small), is the union of constant number of
    these canonical subsets. Proving this requires non-trivial
    modifications of the result of Aronov \textit{et~al.}\xspace
    \cite{aes-ssena-10}. In addition, we show that a measure defined
    over a fat triangle can be covered by a few fat triangles, each
    one of them containing only a constant fraction of the original
    measure. We believe these two results are of independent
    interest. Plugging this into the machinery, previously developed
    for axis parallel rectangles, yields the new approximation
    algorithm. See \secapndref{p:points:in:triangles} for details.

    \item \xparagraph{\Term{P{T}AS}\xspace for disks and planes.} We adapt the
    techniques of Mustafa and Ray \cite{mr-irghs-10} in order to get a
    PTAS for instances consisting of unweighted disks and
    unit-capacity points: we lift the problem to 3d, we construct an
    approximate conflict graph (as done by Mustafa and Ray), and we
    use a local search algorithm. This result also implies a \Term{P{T}AS}\xspace for
    \PackPoints for unweighted points and uniform capacity halfspaces
    in $\Re^3$.  See \secapndref{PTAS} for the details.

    \item \xparagraph{Hardness.} %
    We show some hardness results for our problems. In particular, we
    show that \PackPoints for fat triangles in the plane is as hard as
    independent set in general graphs (see
    \lemref{l:b:pack:pnts:into:triangles}).  We also show that
    \PackRegions is \APXHard (and thus there is no \Term{P{T}AS}\xspace) for
    similarly sized fat triangles in the plane (thus ``matching'' the
    result of \corref{pseudo}).
\end{compactitem}

\paragraph{Main technical contribution.}
Besides the results mentioned above, our work further develops and
extends the techniques for rounding \LP{}s that rise out of low
dimensional geometric constraints. Such work relies on finding the
right order of making decisions about regions as they are being added,
usually initially picking the elements to be considered randomly
according to the value assigned to them by the associated \LP. Such
work in the context of \LP rounding in geometric settings includes
\cite{v-wgscq-10, ch-aamis-09, cch-smcpg-09}.  The basic idea is to
build a conflict graph, on the appropriate random sample, and argue
that there exists a vertex of low degree that can be added without
throwing away too many conflicting vertices.  Our work extends this
approach to more involved settings where conflicts are not just
whether two regions intersect or not (i.e., independent set in a
graph), but rather involve a larger number of regions. To this end, we
prove a combinatorial bound on the expected number of conflicts
realized if we round the associated \LP.  A special easier case of
this was addressed by Chan and Har-Peled \cite{ch-aamis-09} when the
\LP is an independent set \LP. Naturally, in our case the analysis is
considerably more involved.

\paragraph{Previous work.} %
Fox and Pach \cite{fp-cinig-11} presented an $n^{\eps}$ approximation
for independent set for segments in the plane. The usage of \LP
relaxations for approximating such problems is becoming more popular.
In particular, Chalermsook and Chuzhoy \cite{cc-misr-09} use a natural
LP relaxation to get an $O(\log \log m)$-approximation for independent
set of axis parallel rectangles in the plane. The geometric set cover
problem and the more general problem, the geometric set multi-cover
problem, have approximation algorithms that use $\eps$-nets to round
the natural \LP relaxation; see \cite{cch-smcpg-09} and references
therein. Chan and Har-Peled \cite{ch-aamis-09} used local search to
get a \Term{P{T}AS}\xspace for independent set of pseudo-disks.  Independently,
Mustafa and Ray \cite{mr-irghs-10} used similar ideas to get a \Term{P{T}AS}\xspace
for hitting set of pseudo-disks in the plane. There is not much work
on the hardness of optimization problems in the geometric settings we
are interested in.  \cite{cc-ccopd-07} shows that the problem of
independent set of axis-parallel boxes in three dimensions is \APXHard
(the problem is known to be \NPHard in the plane).  See also
\cite{gc-eaahr-11, h-bffne-09} and references therein for some recent
hardness results.  Naturally, in non-geometric settings, there is a
vast literature on the problems and techniques we use, see
\cite{ws-daa-11}. Our algorithms use the randomized rounding with
alteration technique to round a fractional solution. This technique
was used in \cite{s-nacpp-01} to find an approximate solution to
packing integer programs (PIPs) of the form $\{\max w x : Ax \leq b, x
\in \mathbb{Z}_+^{n}\}$, where $A$ is a matrix whose entries are
either $0$ or $1$. The approximation guarantee given in
\cite{s-nacpp-01} is $O(n^{1 / B})$, where $B = \min_i b_i$.

\paragraph{Organization.}
In \secref{prelims} we define the problem and the associated \LP
relaxation, and describe some basic tools used throughout the
paper. In \secref{hypergraph} we present the approximation algorithm
for the hypergraph case. %
\InConfVer{We conclude in \secref{conclusions}. Due to minor space
   limitations, we move many of the technical parts of the paper to
   the appendices. } %
In \secapndref{applications} we present various applications of our
main result.  In \secapndref{p:points:in:triangles} we present the
algorithm for packing points into fat triangles.  In \secapndref{PTAS}
we present a \Term{P{T}AS}\xspace for some restricted cases.  In
\secapndref{hardness} we present some hardness results.  \InFullVer{
   We conclude in \secref{conclusions}.}

\section{Preliminaries}
\seclab{prelims}

For a maximization problem, an algorithm provides an
\emphi{$\alpha$-approximation} if it outputs a solution of value at
least $\Opt/\alpha$, where $\Opt$ is the value of the optimal
solution. An \emphi{$(\alpha, \beta)$-approximation} algorithm for
\PackHGraph is an algorithm that returns a (potentially infeasible)
solution of value at least $\Opt/\alpha$ such that each hyperedge
$\hedge$ contains at most $\max( \capacityX{\hedge}, \beta)$ vertices
of the solution.

\InFullVer{%
\paragraph{$\alpha$-fat triangles.}
For $\alpha \geq 1$, a triangle $\triangle$ is \emphi{$\alpha$-fat} if
the ratio between its longest edge and its height on this edge is
bounded by $\alpha$ (there are several equivalent definitions of this
concept). A set of triangles is $\alpha$-fat if all the triangles in
the set are $\alpha$-fat.  The union complexity of $n$ $\alpha$-fat
triangles is $O(n \log^* n )$ \cite{eas-ibuft-11, abes-ibufop-11} (the
constant in the $O$ depends on $\alpha$, which is assumed to be a
constant).  }

\subsection{\LP Relaxation and the Rounding Scheme}
\seclab{relax}

We consider the following natural \LP relaxation for the \PackHGraph
problem. For each vertex $v$, we have a variable $x_v$ with the
interpretation that $x_v$ is $1$ if $v$ is selected, and $0$
otherwise. For each hyperedge $\hedge$, we have a constraint that
enforces that the number of vertices of $\hedge$ that are selected is
at most the capacity of $\hedge$.

\centerline{%
   \begin{minipage}{0.7\textwidth}%
       \begin{align*}
           \textsc{Hypergraph-LP}\xspace: \qquad \max \quad & \sum_{v \in \VSet} w_v x_v\\
           &\sum_{v \in \hedge} x_v \leq \capacityX{\hedge} \qquad &\forall
           \hedge
           \in \HESet\\
           & 0 \leq x_v \leq 1 \qquad &\forall v \in \VSet.
       \end{align*}
   \end{minipage}%
}

\bigskip
\noindent
The \emphi{energy} of a subset $\VSetA \subseteq \VSet$ is $\ds
\EnergyX{\VSetA} = \sum_{\vertex \in \VSetA} x_\vertex$.  In the
following, $\EuScript{E}$ denotes the \emphi{energy} of the \LP solution;
that is $\EuScript{E} =\EnergyX{\VSet} = \sum_{v \in \VSet} x_v$. Note that
the energy is at most the number of vertices of the hypergraph.  Also,
we assume that $\EuScript{E}\geq 1$ (which is always true since all the
capacities are at least one).

\begin{definition}
    Let $\hgraph = (\VSet, \HESet)$ be a hypergraph. For any integer
    $k$, let $\feq{k}{\cdot}$ denote the function %
    %
    \InFullVer{%
       \begin{align*}
           \feq{k}{t} = \max_{X \subseteq \VSet, |X| \leq t}
           \cardin{\brc{\hedge\sep{ \hedge \in \HESet \text{ and }
                    \cardin{X \cap \hedge} = k+1}}};
       \end{align*}%
    }%
    \InConfVer{%
       $ \feq{k}{t} = \max_{X \subseteq \VSet, |X| \leq t}
       \cardin{\brc{\hedge\sep{ \hedge \in \HESet \text{ and }
                \cardin{X \cap \hedge} = k+1}}}$; }

    that is, $\feq{k}{t}$ is the maximum number of hyperedges of size
    $k+1$ of a sub-hypergraph of $\hgraph$ that is induced by a subset
    of at most $t$ vertices. We say that $\hgraph$ has \emphi{bounded
       growth} if the following conditions are satisfied:
    \begin{compactenum}[~~~(A)]
        \item There exists a non-decreasing function $\gamma(\cdotp)$
        such that $\feq{k}{t} \leq 2^{O(k)} t \gamma(t) $ for any $k$
        and $t$.
        \item There exists a constant $c$ such that $\feq{k}{x t} \leq
        c \feq{k}{t}$ for any $t, k$ and $x$ such that $1 \leq x \leq
        2$.
    \end{compactenum}

    \deflab{bounded}%
\end{definition}

\noindent
This notion of bounded growth is a hereditary property of the
hypergraph, and it is somewhat similar to the bounds on the size of
set systems with bounded \Term{VC}\xspace dimension.  Hypergraphs with bounded
growth arise naturally in geometric settings.

The minimum capacity of a packing instance is a useful measure of how
hard the instance is; formally, the \emphi{minimum capacity} of a
given instance $\hgraph$ is
\begin{equation}
    \nu = \nu(\hgraph) = \min_{\hedge \in \HESet} \capacityX{\hedge}.
    \eqlab{m:c}
\end{equation}

\InFullVer{\subsection{Basic tools}}

Let $\hgraph = (\VSet,\HESet)$ be a hypergraph, and let $\VSetA
\subseteq \VSet$ be a subset of its vertices. The sub-hypergraph of
$\hgraph$ \emphi{induced} by $\VSetA$ is $\InducedX{\hgraph}{\VSetA} =
\pth{\VSetA, \brc{ \hedge \cap \VSetA \sep{ \hedge \in \HESet}}}$.

The following lemma testifies that the packing problem can be solved
in a straightforward fashion if all the capacities are the same (i.e.,
uniform capacities). This is done by repeatedly applying a procedure
to find and remove an independent set in the remaining induced
sub-hypergraph (for example the procedure in \cite{ch-aamis-11}).

\InConfVer{ For some basic tools about these settings, see
   \apndref{basic:tools}.}

\newcommand{\BasicTools}{%
   \begin{lemma}
       Let $\hgraph$ be a hypergraph for which there is a polynomial
       time algorithm \ensuremath{\mathtt{alg}}\xspace that takes as input a fractional solution
       to \textsc{Hypergraph-LP}\xspace for an \PackHGraph instance on $\hgraph$, or any
       induced subgraph of $\hgraph$, with unit capacities --- i.e.,
       an independent set instance --- and it constructs an integral
       solution whose value is at least an $\alpha$ fraction of the
       value of the fractional solution.  Then one can compute in
       polynomial time a $2\alpha$-approximation for any instance of
       \PackHGraph on $\hgraph$ with uniform capacities (i.e., all the
       hyperedges have the same capacity, say $k$).

       \lemlab{basic}
   \end{lemma}

   \begin{proof}
       Let $\hgraph=(\VSet, \HESet)$ be an instance of \PackHGraph in
       which all hyperedges have the same capacity, say $k$. Let
       $\hgraph_0 = \hgraph$, and in the $i$\th iteration, for
       $i=1,\ldots, k$, compute a maximum weight independent set
       $\VSetB_i$ in $\hgraph_{i-1}$ using \ensuremath{\mathtt{alg}}\xspace, and let $\hgraph_{i}
       = \InducedX{\hgraph}{\VSetA \setminus \VSetD_i }$, where
       $\VSetD_i = \VSetB_1\cup \ldots \cup \VSetB_{i}$. We claim that
       $\VSetD_k$ is the required approximation.

       Clearly, no hyperedge of $\hgraph$ contains more than $k$
       vertices of $\VSetD_k$ as it is the union of $k$ independent
       sets, and as such it is a valid solution. Now, let $\VOpt$ be
       the optimal solution. If $w( \VOpt \cap \VSetD_k) \geq
       w(\VOpt)/2$ then we are done. Otherwise, consider the
       hypergraph $\hgraph_{i-1}$, and observe that $\VOpt \setminus
       \VSetD_{i-1}$ is a valid solution for \PackHGraph for this
       graph (with uniform capacities $k$). Interpreting this integral
       solution as a solution to the \LP, and scaling it down by $k$,
       we get a factional solution to the independent set \LP of this
       hypergraph of value $w(\VOpt \setminus \VSetD_{i-1})/k$.  Since
       $\VSetB_i$ was constructed using \ensuremath{\mathtt{alg}}\xspace on the optimal fractional
       solution to the independent set \LP of this hypergraph, we have
       that
       \[
       w(\VSetB_i) \geq \mathrm{opt}_{\text{LP}}(\hgraph_{i-1})/\alpha \geq w( \VOpt
       \setminus \VSetD_{i-1})/k\alpha \geq w( \VOpt \setminus
       \VSetD_{k})/k\alpha \geq w(\VOpt)/2k\alpha.
       \]
       Which implies that $w(\VSetD_k) \geq w(\VOpt)/2\alpha$.
   \end{proof}

   A hypergraph $\hgraph = (\VSet,\HESet)$ \emphi{shatters} $\VSetA
   \subseteq \VSet$ if the number of hyperedges in
   $\InducedX{\hgraph}{\VSetA}$ is $2^{\cardin{\VSetA}}$.  The
   \emphi{\Term{VC}\xspace dimension} of $\hgraph$ is the size of the largest set
   of vertices it shatters.

   The following is a ``sparsification'' lemma. Here we get better
   bounds than the standard technique, as we are using stronger
   sampling results known for spaces with bounded \Term{VC}\xspace dimension.

   \begin{lemma}
       Let $\hgraph = (\VSet, \HESet)$ be an instance of \PackHGraph
       with \Term{VC}\xspace dimension $d$, and consider its fractional \LP
       solution of value $\Opt$ and with energy $\EuScript{E}$. Then, one
       can compute, in polynomial time, a valid fractional solution
       for the \LP of $\hgraph$ such that:
       \begin{compactenum}[(A)]
           \item The value of the new fractional solution is $\geq
           \Opt/12$.
           \item The number of vertices with non-zero value is $O( d
           \EuScript{E} \log \EuScript{E})$.

           \item The value of each non-zero variable is equal to $i/M$
           for some integer $i \leq M$, where $M = O( d \log \EuScript{E}
           )$.

           \item The total energy in the new solution is $\Theta(
           \EuScript{E})$.
       \end{compactenum}

       \lemlab{sparsify}
   \end{lemma}

   \begin{proof

       Let $\eps = 1/\EuScript{E}$, where $\EuScript{E} =\sum_\vertex
       x_\vertex$, and $x_\vertex$ is the value the \LP assigns to
       $\vertex \in \VSet$ in the optimal \LP solution.

       Let $T = O\pth{ d \log \EuScript{E} }$ and $\RSample$ be a random
       sample of $\VSet$ of (expected) size $\tau = \EuScript{E} T = O(
       (d/\eps) \log(1/\eps))$, created by picking each vertex
       $\vertex$ independently with probability $ x_\vertex \cdot T$%
       \footnote{%
          A minor technicality is that $x_\vertex T$ might be larger
          than one. In this case, we put $\floor{ x_\vertex T}$ copies
          of $\vertex$ into $\RSample$, and we put an extra copy of
          $\vertex$ into $\RSample$ with probability $x_\vertex T -
          \floor{x_\vertex T}$. It is straightforward to verify that
          our argumentation goes through in this case. Observe that
          such large values work in our favor by decreasing the
          probability of failure.

       }. %
       This sample is a relative $(\eps, 1/2)$-approximation
       \cite{h-gaa-11, hs-rag-11}, with probability of failure $\leq
       \rho_1 = \eps^{O(d)}$. That implies that for any hyperedge
       $\hedge \in \HESet$ such that $x(\hedge) = \sum_{\vertex \in
          \hedge} x_\vertex$ we have
       \[
       \frac{ \cardin{\RSample \cap \hedge} }{ \cardin{\RSample} }%
       \leq%
       (1+1/2)\pth{ \frac{x(\hedge)}{\EuScript{E}} + \eps }.
       \]
       To interpret the above, observe that $\Ex{\cardin{\RSample \cap
             \hedge} \MakeSBig\!\ts } = x(\hedge) T$ and
       $\Ex{\cardin{\RSample} \MakeSBig\!\ts} = \EuScript{E} T$, as such, a
       rough estimate of the expectation of $\ds \cardin{\RSample \cap
          \hedge} / \cardin{\RSample} $ is ${x(\hedge)} /
       {\EuScript{E}}$. Thus, the above states (somewhat opaquely) that no
       hyperedge is being over-sampled by $\RSample$.

       Since the expected size of $\RSample$ is $\tau$, by Chernoff's
       inequality, we know that $\cardin{\RSample} \leq 2 \tau$ with
       probability at least $1- \rho_2$, where $\rho_2 =
       \eps^{O(d\EuScript{E})}$ (as $\EuScript{E}\geq 1$).  Now, consider a
       hyperedge $\hedge$ with capacity $k$, and observe that
       $x(\hedge) \leq k$. As such, $\cardin{\RSample \cap \hedge}
       \leq (1+1/2)(x(\hedge)/\EuScript{E} + \eps) \cardin{\RSample} \leq
       (3/2) (k+1)\eps 2\tau \leq 6k T$. In particular, if $\vertex$
       appears $t_\vertex$ times in $\RSample$ ($\RSample$ is a
       multiset), then we assign it the fractional value $y_\vertex =
       t_\vertex/6 T$. We then have that $y(\hedge) = \sum_{\vertex
          \in \hedge} y_\vertex \leq \cardin{\RSample \cap \hedge} / 6
       T \leq k$ (and this holds for all hyperedges with probability
       $\geq 1- \rho_1$). As such, the fractional solution defined by
       the $y$'s is valid.

       As for the value of this fractional solution, consider the
       random variable $Z = \sum_{\vertex} y_\vertex w(\vertex)$,
       which is a function of the random sample $\RSample$. Observe
       that
       \[
       \Ex{Z \MakeBig \ts
       =%
       \sum_{\vertex} w(\vertex) \Ex{ \frac{t_\vertex}{6 T} \MakeBig
       }%
       =%
       \sum_{\vertex} \frac{ w(\vertex)}{6 T} \Ex{ t_\vertex \MakeBig
       }%
       =%
       \sum_{\vertex} \frac{ w(\vertex)}{6 T} x_\vertex T

       =%
       \frac{1}{6}\sum_{\vertex} w(\vertex) x_\vertex%
       =%
       \frac{\mathrm{opt}_{\text{LP}}}{6},
       \]
       as $\mathrm{opt}_{\text{LP}} = \sum_{\vertex} w(\vertex) x_\vertex$.  In
       particular, since no vertex can have $w(\vertex) > \mathrm{opt}_{\text{LP}}$
       (otherwise, we would choose it as the solution), it follows
       that
       \[
       Z = \sum_{\vertex} y_\vertex w(\vertex)%
       \leq%
       \mathrm{opt}_{\text{LP}} \sum_{\vertex} \frac{t_\vertex}{ 6 T}%
       \leq%
       \mathrm{opt}_{\text{LP}} \, \frac{\cardin{\RSample}}{ 6 T}%
       \leq%
       \mathrm{opt}_{\text{LP}} \frac{2\tau}{6T}%
       = \mathrm{opt}_{\text{LP}} \frac{2\EuScript{E} T}{6T}%
       \leq%
       \frac{\EuScript{E}}{3}\mathrm{opt}_{\text{LP}}.
       \]
       This implies that $\Prob{ Z \geq \mathrm{opt}_{\text{LP}}/12 \MakeSBig\ts } \geq
       1/4\EuScript{E} = \eps/4$. Indeed, if not,
       \begin{align*}
           \Ex{Z \MakeSBig\ts}%
           &\leq%
           \frac{\mathrm{opt}_{\text{LP}}}{12}\Prob{Z \leq \frac{\mathrm{opt}_{\text{LP}}}{12}}%
           +%
           \Prob{\MakeBig Z \geq \frac{\mathrm{opt}_{\text{LP}}}{12} } \frac{\EuScript{E}}{3}
           \mathrm{opt}_{\text{LP}}%
           \\
           &<%
           \frac{\mathrm{opt}_{\text{LP}}}{12} + \frac{1}{4\EuScript{E}} \cdot
           \frac{\EuScript{E}}{3} \mathrm{opt}_{\text{LP}} %
           \leq%
           \frac{\mathrm{opt}_{\text{LP}}}{6},
       \end{align*}
       a contradiction. As such, a random sample $\RSample$
       corresponds to a valid assignment with value at least
       $\mathrm{opt}_{\text{LP}}/12$ with probability at least $\Prob{ Z \geq \mathrm{opt}_{\text{LP}}/12
       } - \rho_1 - \rho_2 \geq \eps/8$, as $\rho_1 + \rho_2$ is an
       upper bound on the sample $\RSample$ failing to have the
       desired properties. As such, taking $u =O( \EuScript{E} \log
       \EuScript{E})$ independent random samples one of them is the
       required assignment, with probability $\geq 1 - (1-\eps/8)^u
       \geq 1 - 1/\EuScript{E}^{O(1)}$. We take this good sample together
       with its associated \LP values as the desired fractional
       solution to the \LP. Also, note that the total energy of the
       new solution is $\Theta(\EuScript{E})$, since by Chernoff's
       inequality $\tau/2 \leq \cardin{\RSample}\leq 2\tau$ with
       probability at least $1-2\rho_2$.
   \end{proof}
}

\InFullVer{%
   \BasicTools%
}

\section{Approximate packing for hypergraphs}
\seclab{hypergraph}

In this section, we present the algorithm for computing a packing for
a given hypergraph $\hgraph = (\VSet, \HESet)$. We assume that
$\cardin{\HESet}$ is polynomial in $\cardin{\VSet}$ and that $\hgraph$
has the bounded growth property introduced in \defref{bounded}
(properties which both hold for hypergraphs arising out natural
geometric settings). Let $x$ be a solution to the \textsc{Hypergraph-LP}\xspace relaxation
described in \secref{relax}.

\subsection{The algorithm}
\seclab{algorithm}
We round the fractional solution to an integral solution using a
standard randomized rounding with alteration approach. The first step
is to choose an appropriate ordering of the vertices. We will see
later how to choose a good ordering; for now, we assume that we are
given the ordering. The rounding then proceeds in two phases, the
\emphi{selection phase} and the \emphi{alteration phase}. In the
selection phase, we pick a random sample $\RSet$ of the vertices by
picking each vertex $\vertex$ independently at random with
probability $x_{\vertex} / \scl$, where $\scl$ is a parameter that we
will determine later. In the alteration phase, we pick a subset of
$\RSet$ as follows: We consider the sampled vertices in the order
chosen and we add the current vertex to our solution if the resulting
solution remains feasible. We say that a vertex is \emphi{selected}
if it is present in the sample, and we say that it is
\emphi{accepted} if it is present in the solution. The main insight
is that we can take advantage of the bounded growth property of the
hypergraph to show that there is an ordering such that each vertex
is accepted with constant probability, provided that it is selected.
This will immediately imply that the algorithm achieves a
$O(\scl)$-approximation.

\medskip

The main challenge is to prove that a good ordering for the alteration phase exists, that is an ordering such that we accept each selected vertex with constant probability.  We now proceed to give such a proof.  This proof will suggest a natural $O(n^{C+O(1)})$ time brute force algorithm to actually compute this good ordering, where $C$ is the maximum capacity of an edge in the given instance.  In \secref{ordering:polytime} we show how one can remove this exponential dependence on $C$.

\begin{rexample}
    To keep the presentation accessible, we interpret this algorithm
    for instances of \PackRegions in which the regions are disks.
    Specifically, we are given a weighted set of disks
    $\ObjSet$ and set of points $\PntSet$ with capacities. The disks
    of $\ObjSet$ form the set of vertices of the hypergraph and every
    point ${\mathsf{p}} \in \PntSet$ forms a hyperedge; that is the hyperedge
    $\hedge_{\mathsf{p}}$ is the set of all disks of $\ObjSet$ that contain
    ${\mathsf{p}}$.

    In this case, the mysterious quantity $\feq{k}{t}$ (see
    \defref{bounded}) is bounded by the number of faces in an
    arrangement of $t$ disks that are of depth exactly $k+1$.  Since
    the union complexity of $t$ disks is linear, standard application
    of the Clarkson technique implies that $\feq{k}{t} = O(kt)$. In
    particular, for this case $\gamma(t) =O(1)$.
\end{rexample}

\subsubsection{Constructing a good ordering}
Before we describe how to construct a good ordering of the vertices,
it is useful to understand what will force a vertex to be rejected in
the alteration phase. With this goal in mind, consider an ordering of
the vertices. Let $\RSet$ be a sample of the vertices in $\VSet$ such
that each vertex $\vertex$ is in $\RSet$ independently at random with
probability $x_{\vertex} / \scl$. Let $\vertex$ be a vertex in
$\RSet$. When we consider $\vertex$ in the alteration phase, we will
reject $\vertex$ iff there exists a hyperedge $\hedge$ of capacity
$\capacityX{\hedge}$ such that $\hedge$ contains $\vertex$ and we have
already accepted $\capacityX{\hedge}$ vertices of $\hedge$.  The event
that we already accepted $\capacityX{\hedge}$ vertices of $\hedge$ is
difficult to analyze.  However, as we will see, we can settle for a
more conservative analysis that upper bounds the probability that
$\vertex$ is rejected, given that \emph{all} of the vertices in
$\RSet$ that appear before $\vertex$ in the ordering are accepted. (In
the alteration phase, it is possible that not all vertices in $\RSet$
that appear before $\vertex$ will be accepted, but this can only help
us.) Since we are only interested in the event that $\RSet$ contains
$k + 1$ vertices --- the vertex $\vertex$ and $k$ other vertices that
appear before $\vertex$ in the ordering --- that are contained in a
hyperedge of capacity $k$, only the set of vertices that appear before
$\vertex$ in the sample matter, and not the actual ordering of the
vertices. With this observation in mind, we define a
\emphi{$k$-conflict} to be a set of $k + 1$ vertices that are
contained in a hyperedge of capacity $k$.  In the following,
$\conflictSet_k$ denotes the set of all $k$-conflicts, and
$\conflictSet = \cup_k \conflictSet_k$ denotes the set of all
conflicts. We are interested in the probability of the event that all
of the vertices of a $k$-conflict, $\conflict$, are present in the
sample, and we refer to this probability as the \emphi{$\scl$-potential}
of the conflict, $\forceY{\scl}{\conflict}$.  For the analysis it will
also be useful to define the unscaled version of this quantity, that
is the probability that all the vertices of a conflict are present
given that we sampled each vertex with probability $x_\vertex$ and not
$x_\vertex/\scl$.  We refer to this quantity as simply the
\emphi{potential} of the conflict, $\forceX{\conflict}$.
Formally we have,
\[
\forceY{\scl}{\conflict} = \prod_{\vertex \in \conflict}
\frac{x_{\vertex}}{\scl} \qquad \text{ and } \qquad \forceX{\conflict}
= \prod_{\vertex \in \conflict} x_{\vertex}.
\]
Another quantity of interest is the expected number of conflicts in
which a vertex $\vertex$ participates, given that $\vertex$ is in the
sample. We refer to this quantity as the \emphi{$\scl$-resistance} of
a vertex $\vertex$ in a set of vertices $\VSetA \subseteq \VSet$, and
we use $\resistZ{\scl}{\vertex}{ \VSetA}$ to denote it: %
\InFullVer{%
   \[
   \resistZ{\scl}{\vertex}{ \VSetA} %
   =%
   \frac{\scl}{x_{\vertex} } \sum_{\conflict \in \conflictSet,
      \conflict \subseteq \VSetA, \vertex \in \conflict}
   \forceY{\scl}{\conflict}.
   \]%
}%
\InConfVer{%
   $\ds \resistZ{\scl}{\vertex}{ \VSetA} %
   =%
   \frac{\scl}{x_{\vertex} } \sum_{\conflict \in \conflictSet,
      \conflict \subseteq \VSetA, \vertex \in \conflict}
   \forceY{\scl}{\conflict}$.

}%

\paragraph{The ordering.}
Note that, if the $\scl$-resistance of $\vertex$ with respect to the
set $X$ of vertices that come before it in the ordering is small, the
probability of rejecting $\vertex$ is also small. This suggests that
the vertex with least resistance (with respect to $\VSet$) should be
the last vertex in the ordering. This gives us the following algorithm
for constructing an ordering: We compute the vertex of least
resistance and put it last in our ordering (i.e., it is
$\vertex_n$). We then recursively consider the remaining vertices and
we compute an ordering for them. In the following, we assume for
simplicity that the resulting ordering is $\vertex_1,\ldots,
\vertex_n$.

Note that computing the resistance of a vertex by brute force takes
$O\pth{ n^{C+O(1)} }$ time, where $C$ is the maximum capacity of a
hyperedge, and therefore this algorithm is not efficient. We give a
polynomial time algorithm for constructing the ordering in
\secref{ordering:polytime}.

\subsection{Analysis}

Our main insight is that, if the hypergraph satisfies the bounded
growth property defined in \defref{bounded}, then for any set $\VSetA
\subseteq \VSet$ there exists a vertex $\vertex \in \VSetA$ such that
$\resistZ{\scl}{\vertex}{ \VSetA} \leq 1/4$.  We prove this below in
\secref{proof:resistance} (see \lemref{resistance}).  This proof
requires that we set $\scl= \alpha\gamma(\EuScript{E})^{1/\nu}$, where
$\alpha$ is some sufficiently large constant.  As such, in the
remainder of this section we assume $\scl= \alpha\gamma(\EuScript{E})^{1/\nu}$.

We now show that given $\resistZ{\scl}{\vertex}{ \VSetA} \leq 1/4$,
proving the quality of approximation of the algorithm is
straightforward.

\begin{lemma}
    Let $\RSet$ and $\mathcal{O}$ be the set of vertices that were selected and
    accepted by the algorithm, respectively. For each $i$, we have
    $\Prob{\vertex_i \in \mathcal{O} \sep{\vertex_i \in \RSet}} \geq 3/4$.

    \lemlab{prob}
\end{lemma}
\begin{proof:in:appendix}{\lemref{prob}}
	Let $\VSetA_i = \permut{\vertex_1, \ldots,\vertex_i}$. Note that,
	if we selected $\vertex_i$, we rejected $\vertex_i$ in the
	alteration phase only if $\vertex_i$ participates in a conflict
	with some of the vertices in $\{\vertex_1, \ldots, \vertex_{i -
	1}\} \cap \RSet$. Let $Z_i$ be the number of conflicts of
	$\VSetA_i$ that contain $\vertex_i$ and are realized in $\RSet$,
	i.e., $\conflict \subseteq \RSet$. In the following, we show that
	the probability that $Z_i$ is non-zero is at most $1/4$, which
	implies the lemma.

	Consider a $k$-conflict $\conflict = \brc{ \vertex_{j_1}, \ldots,
	\vertex_{j_k}, \vertex_i}$, where each vertex of $\conflict$ is
	in $\VSetA_i$ and $\conflict$ contains $\vertex_i$.  The
	probability that all of the vertices of $\conflict$ are selected,
	given that $\vertex_i$ is selected, is equal to

    $\ds \frac{x_{j_1}}{\scl} \cdot \frac{x_{j_2}}{\scl} \cdots
    \frac{x_{j_k}}{\scl} = \frac{\scl}{x_i} \forceY{\scl}{\conflict}$.

    Therefore we have
    \[
    \Ex{Z_i \sep{ \vertex_i \in \RSet} } =%
    \sum_{\conflict \in \conflictSet, \conflict \subseteq \VSetA_i,
       \vertex_i \in \conflict} \frac{\scl}{x_i} \forceY{\scl}{\conflict}%
    =%
    \resistZ{\scl}{\vertex_i}{ \VSetA_i} \leq \frac{1}{4},
    \]
    where the last inequality follows from \lemref{resistance}
    and the fact that $\vertex_i$ is the vertex of minimum resistance in
    $\VSetA_i$. Thus
    \[
    \Prob{\vertex_i \notin \mathcal{O} \sep{\vertex_i \in \RSet}}%
    \leq%
    \Prob{Z_i > 0 \sep{\vertex_i \in \RSet}} \leq%
    \Ex{Z_i \sep{ \vertex_i \in \RSet} }%
    \leq%
    \frac{1}{4}.
    \]
    Therefore, if $\vertex_i$ is selected, it is accepted with
    probability at least $3/4$.
\end{proof:in:appendix}

\begin{corollary}
	The total expected weight of the set of vertices output by the
	algorithm is $\Omega \pth{\Opt /\gamma(\EuScript{E})^{1/\nu}}$, where
	$\Opt$ is the weight of the optimal solution, and $\nu$ is the
	minimum capacity of the given instance.

    \corlab{result}
\end{corollary}

\begin{proof:in:appendix}{\corref{result}}
    By \lemref{prob}, for each vertex $\vertex \in \VSet$, we have
    \begin{align*}
        \Prob{ \mbb \vertex \in \mathcal{O}}%
        &=%
        \Prob{ \mbb \MakeSBig \pth{\vertex \in \mathcal{O}} \cap \pth{
		\vertex
              \in \RSet}}%
        =%
        \Prob{\vertex \in \mathcal{O} \sep{\vertex \in \RSet}} \Prob{ \mbb
		\vertex
           \in \RSet}%
        \geq%
        \frac{3}{4} \Prob{ \MakeSBig \vertex \in \RSet}\\
        &\geq%
        \frac{3x_{\vertex}}{4 \scl},
    \end{align*}
	where $\scl = O \pth{ \gamma(\EuScript{E})^{1/\nu} }$.  By linearity
	of expectation, we have that the expected weight of the generated
	solution is at least
    \[
    \sum_{\vertex \in \VSet} \frac{3x_{\vertex}}{4 \scl} w_{\vertex}%
    =%
    \Omega\pth{ \frac{\sum_{\vertex} x_{\vertex}
          w_{\vertex}}{\gamma(\EuScript{E})^{1/\nu}}}%
    =%
    \Omega\pth{ \frac{\Opt}{\gamma(\EuScript{E})^{1/\nu}}},
    \]
    as $\sum_{\vertex} x_{\vertex} w_{\vertex}$ is the value of the
    fractional \LP solution, which is bigger than (or equal to) the
    weight of the optimal solution.
\end{proof:in:appendix}

\subsubsection{On the expected number of conflicts being realized}

To analyze the algorithm we need to understand how conflicts might
form during its execution, and show that the damage of such conflicts
to the generated solution is limited. To this end, consider the
quantity
\begin{align*}
    \feq{k}{t} = \max_{A \subseteq \VSetA, |A| \leq t}
    \cardin{\brc{\hedge\sep{ \hedge \in \HESet \text{ and } \cardin{A
                \cap \hedge} = k+1}}}.
\end{align*}
This is the maximum number of $k$-conflicts that can be realized for a
set of $t$ vertices.  The quantity of interest in the following is $
\sum_{\conflict \in \conflictSet_k} \forceX{\conflict}$, as it is the
expected number of conflicts that would be realized if we sample
according to the \LP solution. Our purpose is to prove that this
quantity is bounded by a function of the energy of the \LP (the bound
would involve the function $\feq{k}{\cdot}$ defined above).

With this goal in mind, we let $\RSample$ be a random sample of
$\VSetA$ such that each vertex $\vertex \in \VSetA$ is in $\RSample$
independently at random with probability $x_{\vertex} / 2$. We stress
that $\RSample$ is a random sample that we use for the purposes of
defining a quantity $\EuScript{M}$ (i.e., the expected number of conflicts
realized in $\RSample$), and it should not be confused with the random
sample $\RSet$ that is used by the algorithm. In the following, we
bound $\EuScript{M}$ from above in \lemref{upper-bound} and from below in
\lemref{lower-bound}. Putting these two bounds together imply the
desired bound on $\sum_{\conflict \in \conflictSet_k}
\forceX{\conflict}$.

A conflict $\conflict \in \conflictSet$ is \emphi{realized} in
$\RSample$ if there is a hyperedge $\hedge \in \HESet$ such that $
\conflict = \hedge \cap \RSample$ and $\cardin{\conflict} =
\capacityX{\hedge} + 1$.

The following is similar in spirit to the Clarkson technique (a
similar but simpler argument was used by Chan and Har-Peled
\cite{ch-aamis-11}).

\begin{lemma}
    The expected number of $k$-conflicts realized in $\RSample$ is
    $\EuScript{M} = O(\feq{k}{\EuScript{E}(X)})$, where $\RSample$ is a random
    sample of $\VSetA$ such that each vertex $\vertex \in \VSetA$ is
    in $\RSample$ independently at random with probability
    $x_{\vertex} / 2$.

    \lemlab{upper-bound}
\end{lemma}
\begin{proof:in:appendix}{\lemref{upper-bound}}
    Each $k$-conflict $\conflict$ that is realized corresponds to a
    hyperedge $\hedge$ with capacity $k$ such that $\conflict = \hedge
    \cap \RSample$. Additionally, two realized conflicts that are
    distinct correspond to different hyperedges.  Therefore the number
    of $k$-conflicts that are realized in $\RSample$ is at most the
    number of hyperedges $\hedge$ such that the capacity of $\hedge$
    is $k$ and $|\hedge \cap \RSample| = k + 1$.  It follows from the
    definition of $\feq{k}{\cdotp}$ that the number of $k$-conflicts
    is at most $\feq{k}{|\RSample|}$. Therefore it suffices to upper
    bound the expected value of $\feq{k}{|\RSample|}$.

    Note that $\Ex{\cardin{\RSample}} = \EuScript{E}(\VSetA)/2$. We have
    \begin{align*}
        \Ex{ \feq{k}{|\RSample|} \MakeBig\! }%
        &\leq%
        \sum_{t=0}^\infty \Prob{\cardin{\RSample} \geq t
           \frac{\EuScript{E}(\VSetA)}{2}} \feq{k}{ \MakeSBig
           (t+1)\frac{\EuScript{E}(\VSetA)}{2}}%
        \leq%
        \sum_{t=0}^\infty 2^{-(t+1)/2} \feq{k}{ \MakeSBig
           (t+1)\frac{\EuScript{E}(\VSetA)}{2}}%
        \\
        &\leq%
        \sum_{t=0}^\infty 2^{-(t+1)/2} c^{O( \log t)} \, \feq{k}{
           \MakeSBig \EuScript{E}(\VSetA)}%
        =%
        O\pth{ \MakeBig \feq{k}{ \EuScript{E}(\VSetA) } },
    \end{align*}
    since $\hgraph$ has the bounded growth property (see
    \defref{bounded}), and by the Chernoff inequality (we use here
    implicitly that $\EuScript{E}(\VSetA) \geq 1$).
\end{proof:in:appendix}

\begin{lemma}
    For each $k$-conflict $\conflict$, the probability that
    $\conflict$ is realized in $\RSample$ is at least
    $\forceX{\conflict} / 2(2e)^{k}$. Therefore the expected number of
    $k$-conflicts realized in $\RSample$ is $\EuScript{M} =
    \Omega\left(\left(\sum_{\conflict \in \conflictSet_k, \conflict
               \subseteq X} \forceX{\conflict}\right) /
        (2e)^k\right)$.

    \lemlab{lower-bound}
\end{lemma}
\begin{proof:in:appendix}{\lemref{lower-bound}}
    Let $\hedge \in \HESet$ be a hyperedge with capacity $k$ that
    generated the conflict $\conflict$. Since $x$ is a feasible
    solution for the \LP, we have that $\sum_{\vertex \in \hedge -
       \conflict} x_{\vertex} \leq \sum_{\vertex \in \hedge}
    x_{\vertex} \leq \capacityX{\hedge} = k$.  Clearly, the conflict
    $\conflict$ is realized if we pick all the vertices of
    $\conflict$, and none of the vertices of $\hedge - \conflict$, and
    the probability of that event is
    \begin{align*}
        \prod_{\vertex \in \conflict} \frac{x_{\vertex}}{2}
        \prod_{\vertex \in \hedge - \conflict} \pth{1 -
           \frac{x_{\vertex}}{2}} &\geq%
        \frac{1}{2^{k+1}} \prod_{\vertex \in \conflict} x_{\vertex}
        \prod_{\vertex \in \hedge - \conflict} \exp \pth{ -
           x_{\vertex}}%
        \\
        &=%
        \frac{\forceX{\conflict}}{2^{k+1}} \cdot \exp \pth{ -
           \sum_{\vertex \in \hedge - \conflict} x_{\vertex}}%
        \geq%
        \frac{\forceX{\conflict}}{2(2e)^k},
    \end{align*}
    In the first line we used the inequality $1 - x_{\vertex} / 2 \geq
    \exp(-x_{\vertex})$, which holds since $x_{\vertex} \leq 1$.
\end{proof:in:appendix}

Putting the above two lemmas together, we get the following.

\begin{lemma}
    For any non-negative integer $k$ we have $\ds\sum_{h \in
       \conflictSet_k, h \subseteq \VSetA} \forceX{\conflict} =
    O\pth{(2e)^{k} \feq{k}{\EuScript{E}(\VSetA)} \MakeBig\! }$.

    \lemlab{total-energy}
\end{lemma}

\begin{rexample}
    In our running example, we have that the expected number of
    $k$-conflicts that are being realized by a random sample (sampling
    more or less according to the \LP values) is $ \sum_{\conflict \in
       \conflictSet_k} \forceX{\conflict} = O\pth{(2e)^{k} k\EuScript{E}
       \MakeBig\! }$. This is a hefty quantity,
    but the key observation is that if we sample according to the \LP
    values scaled down by a large enough constant, then the
    probability of such a conflict to be realized drops exponentially
    with $k$. In particular, for a sufficiently large constant, the
    expected number of realized $k$-conflicts in such a smple is going
    to be $\leq \EuScript{E}/\pth[]{10 \cdot 2^k}$. Intuitively, this
    implies that such conflicts can only cause the algorithm to drop
    very few vertices during the rounding stage, thus guaranteeing a
    good solution.
\end{rexample}

\subsubsection{Resistance is futile, if you pick the right vertex}

\seclab{proof:resistance}

In the following, we consider a subset $\VSetA$ of the vertices and we
show that there exists a vertex $\vertex \in \VSetA$ whose
$\scl$-resistance $\resistZ{\scl}{\vertex}{\VSetA}$ is at most $1/4$.
Recall that $\conflictSet_k$ is the set of all $k$-conflicts involving
vertices in $\VSet$. We can rewrite the $\scl$-resistance of $\vertex$
in $\VSetA$ as
\begin{align*}
    \resistZ{\scl}{\vertex}{\VSetA} =
    \frac{\scl}{x_{\vertex} } \sum_{\conflict \in \conflictSet, \conflict
       \subseteq \VSetA, \vertex \in \conflict}
       \forceY{\scl}{\conflict} =
    \frac{1}{x_{\vertex}} \sum_{k =
       \nu}^{\infty} \frac{1}{\scl^k} \sum_{\conflict \in
       \conflictSet_k, h \subseteq \VSetA, \vertex \in \conflict}
    \forceX{\conflict}.
\end{align*}
As shown in \lemref{total-energy}, we can relate the total potential
of the conflicts of $\conflictSet_k$ that are contained in $\VSetA$ to
the maximum number of $k$-conflicts contained in a set of at most
$\EuScript{E}(\VSetA)$ vertices, where $\EuScript{E}(\VSetA) = \sum_{\vertex \in
   \VSetA} x_{\vertex}$.

Recall that the hypergraph has the bounded growth
property (see \defref{bounded}) and this property is
hereditary. Therefore the function $\feq{k}{\cdotp}$ in the lemma
above has the two properties described in \defref{bounded} and we get
the following corollary.

\begin{corollary}
    We have $\sum_{h \in \conflictSet_k, h \subseteq \VSetA}
    \forceX{\conflict} = O\pth{2^{ck} \EuScript{E}(\VSetA)
       \gamma(\EuScript{E}(\VSetA)})$, where $c$ is a constant.

    \corlab{total-energy}
\end{corollary}

We can use \corref{total-energy} to complete the
proof of \lemref{resistance} as follows.

\begin{lemma}
    Suppose that the hypergraph $\hgraph$ satisfies the bounded growth
    property (see \defref{bounded}). Let $\scl = \alpha\,
    \gamma(\EuScript{E})^{1/\nu}$, where $\alpha > 0$ is a sufficiently
    large constant and $\nu$ is the minimum capacity of the given
    instance (see \Eqref{m:c}).  Then, for any set $\VSetA \subseteq
    \VSet$, there exists a vertex $\vertex \in \VSetA$ such that
    $\resistZ{\scl}{\vertex}{ \VSetA} \leq 1/4$.

    \lemlab{resistance}
\end{lemma}

\begin{proof}
    Let $T = \sum_{\vertex \in \VSetA} x_{\vertex}
    \resistZ{\scl}{\vertex}{\VSetA}$. The quantity $T /
    \EuScript{E}(\VSetA)$ is the weighted average of the resistances of the
    vertices in $\VSetA$, where the weight of a vertex $\vertex$ is
    $x_{\vertex} / \EuScript{E}(\VSetA)$.  Therefore it suffices to show
    that $T \leq \EuScript{E}(\VSetA) / 4$, since the minimum resistance is
    at most the weighted average. We have

    \begin{align*}
        T &=%
        \sum_{k=\nu}^\infty \frac{1}{\scl^k} \sum_{\vertex \in \VSetA}
        \sum_{\substack{\conflict \in \conflictSet_k\\
              \conflict \subseteq
              \VSetA\\
              \vertex \in \conflict}} \forceX{\conflict} =%
        \sum_{k=\nu}^\infty \frac{k + 1}{\scl^k}
        \sum_{\substack{\conflict \in \conflictSet_k\\
              \conflict \subseteq \VSetA }} \forceX{\conflict} =%
        \sum_{k=\nu}^\infty
        \frac{k+1}{\scl^k} \sum_{\substack{\conflict \in \conflictSet_k\\
              \conflict \subseteq \VSetA}} \forceX{\conflict}%
        \\ &=%
        \sum_{k=\nu}^\infty \frac{k+1}{\scl^k} O\pth{ 2^{ck}
           \EuScript{E}(\VSetA) \gamma(\EuScript{E}(\VSetA)) \MakeBig\!}

        \leq %
        \EuScript{E}(\VSetA) \cdot \underbrace{\beta \sum_{k=\nu}^\infty {
              \pth{\frac{2^c}{\scl}}^k {} (k+1)
              \gamma(\EuScript{E}(\VSetA)})}_{=S},

    \end{align*}
    by \corref{total-energy}, where $\beta$ is some constant.  Since
    $\scl = \alpha\gamma(\EuScript{E})^{1/\nu}$, we have
    \begin{align*}
        S &=%
        { \sum_{k=\nu}^\infty \beta \pth{\frac{2^c}{\alpha}}^k {} (k+1)
           \pth{ \frac{1}{\gamma(\EuScript{E})}}^{k/\nu}
           \gamma(\EuScript{E}(\VSetA))}%
        \leq%
        { \sum_{k=\nu}^\infty \beta \pth{\frac{2^c}{\alpha}}^k {} (k+1)
           \pth{ \frac{1}{\gamma(\EuScript{E})}}^{k/\nu} \gamma(\EuScript{E})}%
        \leq%
        \frac{1}{4},
    \end{align*}
    In the second to last inequality, we have used the fact that
    $\gamma(\cdotp)$ is non-decreasing. The last inequality follows if
    we pick $\alpha$ to be a sufficiently large constant.  Therefore $T
    \leq \EuScript{E}(\VSetA) / 4$, and the lemma follows.
\end{proof}

\newcommand{\ImproveRunningTime}{%
\subsection{Improving the running time}
\seclab{ordering:polytime}%
\apndlab{improve:r:t}%

In \secref{hypergraph}, we described an algorithm that
constructs an ordering of the vertices by repeatedly finding the
vertex of least resistance with respect to the set of remaining
vertices.  Computing the resistance of a vertex by brute force takes
$O(n^{C+O(1)})$ time, where $C$ is the maximum capacity of an edge in
$\HESet$.  However, for our analysis to go through, we only need to
find a vertex that is safe with respect to the set of remaining
vertices; informally, a vertex $\vertex$ is safe if the probability
that it participates in a conflict with a random
sample of the remaining vertices is smaller than some constant (that
is strictly smaller than one), where each remaining vertex $u$ is
included in the sample with probability $x_u / \scl$.  In this section
we show that there is a sampling algorithm that finds a safe vertex
with high probability and its running time is polynomial in the
maximum capacity $C$.

\begin{lemma}
	Computing a good ordering of the vertices can be done in
	polynomial time. Namely, the algorithm of \secref{hypergraph} can
	be implemented in polynomial time.
\end{lemma}

\begin{proof}
    To get the same quality of approximation we do not need to take
    the vertex of least resistance in each round (of computing the
    ordering), but merely a vertex that is ``safe.'' More precisely,
    let $\VSetA$ be the current set of vertices, let $\vertex$ be a
    vertex of this set, and let $\RSample$ be a random sample of
    $\VSetA$ in which each vertex $u$ is included with probability
    $x_u / \scl$ (also we force $\vertex$ to be in $\RSample$).  We
    say that $\vertex$ is \emphi{violated} in $\RSample$ if $\vertex$
    is contained in a hyperedge $\hedge$ such that the number of
    vertices of $\hedge$ that are in $\RSample$ is larger than its
    capacity $\capacityX{\hedge}$. Let $\mu(\vertex, \VSetA)$ denote
    the probability that $\vertex$ is violated in $\RSample$. Note
    that $\mu(\vertex, \VSetA)$ is a (conservative) upper bound on the
    probability that $\vertex$ is rejected by our rounding algorithm
    if we started with an ordering in which $\VSetA\setminus
    \{\vertex\}$ is the set of all vertices that come before
    $\vertex$. Therefore, in order for our rounding to succeed, in
    each round we only need to find a vertex $\vertex$ for which the
    probability $\mu(\vertex, \VSetA)$ is low, where $\VSetA$ is the
    set of all vertices that still need to be ordered at the beginning
    of the round. (We remark that it follows from the argument of
    \lemref{resistance} that, for any set $\VSetA$, there is a vertex
    $\vertex$ for which $\mu(\vertex, \VSetA) \leq 1/4$.)

    Now we are ready to describe how to construct an ordering for our
    algorithm. Let $\VSetA$ be the set of vertices that still need to
    be ordered. As we will see shortly, for each vertex $\vertex \in
    \VSetA$, we can compute an estimate $\overline{\mu}(\vertex,
    \VSetA )$ of the probability $\mu(\vertex, \VSetA )$. We pick the
    vertex $\vertex$ with minimum estimated probability
    $\overline{\mu}(\vertex, \VSetA)$, we make $\vertex$ the last
    vertex (in the ordering of $\VSetA$) and we recursively order
    $\VSetA \setminus \{\vertex\}$.

    We can compute the estimates $\overline{\mu}(\vertex, \VSetA)$ in
    polynomial time as follows. Fix a vertex $\vertex$. Let $\Pnum$ be
    a sufficiently large polynomial in $n$. We pick $\Pnum$
    independent random samples of $\VSetA$ (again, forcing $\vertex$
    to be in each of these samples); in each random sample, each
    vertex $u$ is included with probability $x_u / \scl$. We set
    $\overline{\mu}(\vertex, \VSetA )$ to be the fraction of the
    samples in which the vertex $\vertex$ is violated. Using a
    standard argument based on the Chernoff inequality, we can show
    that our estimates are very close with high probability, and
    therefore our rounding algorithm achieves the required
    approximation with high probability as well; we omit the easy but
    tedious details.
\end{proof}
}

\InConfVer{See \apndref{improve:r:t} for details of how to improve the
   running time.}

\InFullVer{\ImproveRunningTime}

\subsection{The result}
\begin{theorem}
    Let $\hgraph = (\VSet, \HESet)$ be a hypergraph with a weight
    function $\weight{\cdot}$ on the vertices and a capacity function
    $\capacityX{\cdot}$ on the edges, such that $\cardin{\HESet}$ is
    polynomial in $\cardin{\VSet}$ and $\hgraph$ satisfies the bounded
    growth property (see \defref{bounded}); that is, $\feq{k}{t} =
    2^{O(k)} t \gamma(t)$. Then we can compute in polynomial time a
    subset $X \subseteq \VSet$ of vertices such that no hyperedge
    $\hedge$ contains more than its capacity $\capacityX{\hedge}$
    vertices of $X$.  Furthermore, in expectation, the total weight of
    the output set is $\Omega \pth{\Opt /\gamma(\EuScript{E})^{1/\nu}}$,
    where $\Opt$ is the weight of the optimal solution, and $\nu$ is
    the minimum capacity of the given instance.

    \thmlab{p:hypergraph}
\end{theorem}

\InConfVer{
\section{Conclusions}
\seclab{conclusions}%

In this paper, we presented a general framework for approximating
geometric packing problems with non-uniform constraints. We then
applied this framework in a systematic fashion to get improved
algorithms for specific instances of this problem, many of which
required additional non-trivial ideas. There are several special cases
of this problem for which we currently do not know any useful
approximation; for example, the special case of packing axis-parallel
boxes into points, in which the boxes are in four dimensions is still
wide open. Making some progress on these special cases is an
interesting direction for future work.

\section*{Acknowledgments}

The authors thank Timothy Chan, Chandra Chekuri, and Esther Ezra for
several useful discussions.
}

\InConfVer{%
\bibliographystyle{alpha}%

redpajama_set_name: RedPajamaArXiv


text:
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Whidbey's SeaWatch Estate Drops $6M From Price Tag
Phenomenal views, built to live within its environment, this 25 acre Whidbey estate didn't sell at $10M. Will it be snapped up at $4M?
By Tom Trimbath@tetrimbath Jun 1, 2016, 9:15am PDT
Back in April 2015, just about 14 months ago, we wrote about SeaWatch, a $10M waterfront estate on Whidbey Island with a marvelous location and views . Now, it is back on the market at $3,990,000. Something must have changed to drop the price 60 percent. Let's scroll through the numbers.
Then, it was listed as 25 acres, now ~3 acres. That could explain it, but the listing does mention two parcels, including 22 acres of tidelands getting the total back up to 25.
Then it was 2 bedrooms and 3.75 baths; yes, a $10M property with only 2 bedrooms - but very nice ones. That hasn't changed. The 4,349 square footage hasn't changed, either.
The property has an impressive 365 feet of high-bank waterfront, a separate building that can act as an artist's studio or a 20 person mini-conference center, and the estate has managed to accommodate 200 on occasion.
Maybe it was simply a matter of pricing. When it was at $10M, it sat on the market for over seven months. Maybe even event spaces built to work with the environment on a unique site have difficulty commanding seven digits. This will be one to watch. How quickly will it sell after the price was dropped 60%?
· 6675 Wahl Rd, Freeland [Estately]
· Sea Watch [SW]
· Huge Freeland Estate Certainly Not Free [CS]
The Seattle filming locations of '10 Things I Hate About You,' mapped No, there are no sheep
The 25 best things to do in Seattle with kids From beaches that will make you feel like you're in California to top-notch theater productions geared towards the younger set, the Emerald City is full of things to do as a family.


text:
Q: ibm blockchain platform [[{ code: 71, message: 'Authorization failure' }]] I am trying to register a user in a blockchain network that is using hyperledger fabric in its version 2.5.3, I was able to enroll the administrator user, I was able to send data, but when registering a user it shows me the following error
[[ { code: 71, message: 'Authorization failure' } ]]
The error occurs when trying to make the following register.
const secret = await caClient.register(
      {
        enrollmentID: "user",
        role: "client",
      },
      adminUser
    );

I already validated the authentication data of the administrator user and they are correct, I am using the ibp tool to create the hypereldger network.
To register the user I am using the latest examples from the official hyperledger repository

A: Is your client connection profile file up to date i.e. is it correct for the latest running instance of the blockchain network. You will find that creating an admin will nearly always work, but the register user requires the ability to connect to the network. This has been my experience, so be absolutely sure that the current credentials are correct.

redpajama_set_name: RedPajamaStackExchange


International Review of Cytology presents current advances and comprehensive reviews in cell biology - both plant and animal. Authored by some of the foremost scientists in the field, each volume provides up-to-date information and directions for future research. Articles in this volume address endogenous ligands of PACAP, VIP receptors in the autocrine-paracrine regulation of the adrenal gland; ultrastructural dynamcs of human reproduction, from ovulation to fertilization and early embryo development; chromosomal variation in mammalian neuronal cells; automated interpretation of protein subcellular location patterns; cell and molecular biology of human lacrimal gland and nasolacrimal duct mucins.
Endogenous Ligands of PACAP, VIP Receptors in the Autocrine-Paracrine Regulation of the Adrenal Gland.
Ultrastructural Dynamics of Human Reproduction, from Ovulation to Fertilization and Early Embryo Development.
Chromosomal Variation in Mammalian Neuronal Cells: Known Facts and Attractive Hypotheses.
Automated Interpretation of Protein Subcellular Location Patterns.
Cell and Molecular Biology of Human Lacrimal Gland and Nasolacrimal Duct Mucins.
Kwang Jeon received his Ph.D. in cell physiology at King's College, University of London, UK, in 1964 and taught at SUNY Buffalo and University of Tennessee. His research was concerned with the biogenesis and function of cell components in two major areas: Integration of intracellular symbionts into host cells leading to the acquisition of new cell components and cell variation; Membrane-protein recycling during endo- and exocytosis.


This is such a book, first released one year ago in October 2012 and reissued now just when it seems a thawing of 33 years of enmity has begun between the United States and the Islamic Republic of Iran. This last chance at peace was anticipated in its pages. Will the birth of hope for that peace dissolve at last its lurking shadow of horror: a war foreseen in the utterances of prophets too horrible to fulfill?
John Hogue, defines himself as a "Rogue" Scholar. He is a world renowned authority on Nostradamus and other prophetic traditions. His work is critically acclaimed for cutting through the social, religious, and nationalistic projections of prophets and their interpreters to find threads of insight they all share.
Futurist, accurate forecaster in his own right, creator of what some people call the Huffington Post of prophecy websites, HogueProphecy.com, Hogue has received wide international recognition appearing on over 100 television and 1,000 blog and talk radio shows on five continents: North and South America, Asia, Australia and Europe. At present, he has the most multimedia exposure of any prophecy scholar in history.
Hogue has published 23 books in 19 languages. He currently lives on an island in the Pacific Northwest.
I have the PDF version of this book and found it to be very detailed and informative; it helped my understanding of the different Middle Eastern players, a subject that almost demands its own score card, especially for Western Readers.
I am really looking forward to the new Kindle version update because John is always very careful to update, change or even shred his previous speculations when the facts warrant it and/or to explain how things may be appearing one way on the surface but actually playing out very differently behind the scenes.
This is one of John's more complex books, because Middle Eastern Politics and Nostradamus are both tricky and complex subjects. I found the best way to read it was in chunks, followed by some hot tea and time to digest them; them off to read more chunks. I usually read books all the way through at once, but this one has so much information I think the slower method works better.
Rated five stars and all well deserved!
John Hogue is an interesting and entertaining writer. I read his prophecies and check them out. He is a visionary. He is accurate. He also understands how the stars, Zodiac and solar system affect us or even, run us and control us. He knows what happens in the Middle East is of great importance to everyone on the planet. I look forward to his next book of prophecy.
John Hogue has a way with words that keeps you interested and, in my case, expands my perception of the events happening around us. His vast knowledge and discernment concerning history and astrology blended with his sharp intuition makes this book and his other writings very special.
3.0 out of 5 starsGood political commentary, but in some ways confusing.
This book is a good political commentary on American relations with Iran and the Middle East in general and its intuitive extrapolations as to what could happen are pretty much on the mark. Prophesy may not be the important part here as Hogue's political intuition seems more important. Also, the book keeps switching around to different subjects, some personal and other matters which can confuse the reader. But I would recommend reading this book to anyone concerned about current events and their direction.
I like anything written by John Hogue and follow him on facebook and coast to coast radio.
Not earth shaking, lol, but another good observation from our dear John, no pun intended.


At the first plenary session of the Northern Caribbean University (NCU) board held at the Wyndham Resort in Kingston, Jamaica, Mr. Keith Major, a local elder and member of the Johnson Park Seventh-day Adventist Church in the South Bahamas Conference, was one of two additional members appointed on June 7, 2011, to serve on the university board for the next five years.
Keith Major, who is the host for ACUM's weekly television broadcast, Voices, and an executive committee member of ACUM, is well-known in the local Bahamian community. He is a former chairman of the Bahamas Electricity Corporation and has a 30-year career in the insurance industry, having served as a vice-president of three companies. A motivational speaker in demand, he received an honorary doctorate from Sojourner-Douglass College, Baltimore, Maryland and is currently the president and CEO of Kits Ltd, a training company.
Mr. Major is an alumnus of West Indies College (now Northern Caribbean University) and is married to Linda nee Lewis. They have three boys and one granddaughter. When told of his appointment, Keith said, "I am happy to have been selected to such a high honor of serving on this most prestigious school board. I am humbled that they chose me and thankful that my passion for learning and teaching has found a higher level where my influence can be felt."


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