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\begin{document}

\title{\bf Approximability of the Minimum Weighted Doubly Resolving Set Problem \thanks{Research supported in part by NNSF of China under Grant No. 11222109, 11021161 and 10928102,
by 973 Project of China under Grant No. 2011CB80800,   and by CAS Program for Cross \& Cooperative Team of Science  \& Technology Innovation.}}

 \author{Xujin Chen \and Xiaodong Hu \and  Changjun Wang}

\date{Academy of Mathematics and Systems Science \\ Chinese Academy
of Sciences, Beijing 100190, China\\
${}$\\
\mailsa}

\maketitle

\begin{abstract} {Locating source of diffusion in networks is crucial for controlling and preventing epidemic   risks. It has been studied under various probabilistic models. In this paper, we study source location  from a deterministic point of view by modeling it as the minimum weighted doubly resolving set (DRS) problem, which is a strengthening of the well-known metric dimension problem.}

 \hspace{4mm}Let $G$ be a vertex weighted undirected graph on $n$ vertices. A vertex subset $S$ of $G$ is  DRS  of $G$ if for every pair of vertices $u,v$ in $G$, there exist $x,y\in S$ such that the difference of distances (in terms of number of edges) between $u$ and $x,y$ is {not equal to} the difference of distances  between $v$ and $x,y$. The minimum weighted DRS problem consists of finding a DRS in $G$  with  minimum total weight.
 We establish $\Theta(\ln n)$ approximability of the minimum DRS problem on general graphs for both weighted and unweighted versions. This is the first work providing explicit approximation lower and upper bounds for minimum (weighted) DRS problem, which are nearly tight. Moreover, we design first known strongly polynomial time algorithms for the minimum weighted DRS problem on   general wheels and trees with additional constant $k\ge0$ edges.
\end{abstract}

\noindent{\bf Keywords:} {Source location, Doubly resolving set, Approximation  algorithms, Polynomial-time solvability, Metric dimension

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\section{Introduction}

{Locating the source of a diffusion in  complex networks is an intriguing challenge, and finds diverse applications in controlling and preventing network epidemic risks \cite{sz11}. In particular, it is often financially and technically
impossible to observe the state of all vertices in a large-scale network, and, on the other hand, it
  is desirable to find the location of the source (who initiates the diffusion) from measurements collected by sparsely placed {observers} \cite{ptv12}. Placing an observer at vertex $v$ incurs a cost, and the observer with a clock   can record the time at which the state of $v$ is changed (e.g.,
knowing a rumor, being infected or contaminated).
 Typically, the time when the {single} source originates an information is unknown \cite{ptv12}. The observers  can only report  the times they receive the information, but the senders of the information (i.e., we do not know who infects whom or who influences whom) \cite{glk10}. The information is diffused from the source to any vertex through shortest paths in the network, i.e., as soon as a vertex receives the information, it sends the information to all its neighbors simultaneously, which takes one time unit.
Our goal is to select a subset $S$ of vertices with minimum total cost {such that}
the source can be uniquely located by the ``infected" times of vertices in $S$. This
problem is equivalent to finding a minimum weighted {\em doubly resolving set} (DRS) in networks defined as follows.}

 \paragraph{DRS model.}  {Networks are modeled as undirected connected graphs without parallel edges nor loops}.  Let $G=(V,E)$ be a     graph on $n\ge2$ vertices, and each vertex $v\in V$ has a nonnegative {\em weight} $w(v)$, {representing its cost}. For any $S\subseteq V$, the {\em weight} of $S$ is defined to be $w(S):=\sum_{v\in S}w(v)$. For any $u,v\in V$, we use $d_G(u,v)$ to denote the {\em distance} between $u$ and $v$ {in $G$}, i.e., the number of edges in a  shortest path between $u$ and $v$. Let $u,v,x,y$ be four distinct vertices of $G$. Following C\'aceres et al. \cite{cjmppsw07}, we say that $\{u,v\}$ {\em doubly resolves} $\{x,y\}$, or $\{u,v\}$ {\em doubly resolves} $x$ and $y$, if \[d_G(u,x)-d_G(u,y)\ne d_G(v,x)-d_G(v,y).\] Clearly $\{u,v\}$ doubly resolves $\{x,y\}$ if and only if $\{x,y\}$ doubly resolves $\{u,v\}$. {For any subsets $S,T $ of vertices,  $S$} {\em doubly resolves} $T$ if every pair of vertices in $T$ is doubly resolved by some pair of vertices in $S$. {In particular, $S$ is called} a {\em doubly resolving set} (DRS) of $G$ if $S$ doubly \mbox{resolves $V$.} Trivially, $V$ is a \mbox{DRS of $G$.} The {\em minimum weighted doubly resolving set} (MWDRS) problem is to find a DRS of $G$ that has a minimum weight {(i.e. a minimum weighted DRS of $G$)}. In the special case where all vertex weights are equal to $1$, the problem is referred to as the {\em minimum doubly resolving set} (MDRS) problem {\cite{mkkm12}},  and it concerns with the minimum cardinality ${\tt dr}(G)$ of    DRS of $G$

 {Consider arbitrary $S\subseteq V$. It is easy to see that   $S$ fails to locate the diffusion source in $G$} at some case if and only if there exist distinct vertices $u,v\in V$ such that $S$ cannot distinguish between the case of $u$ being the source and that of $v$ being the source, i.e., $d_G(u,x)-d_G(u,y)=d_G(v,x)-d_G(v,y)$ for any $x,y\in S$; equivalently, $S$ is not a DRS of $G$. \bluecomment{(See Appendix \ref{apx:appl}.)} Hence, the MWDRS problem models exactly the problem of finding cost-effective observer {placements} for locating source, as mentioned in our opening paragraph.

\paragraph{Related work.}
 {Epidemic diffusion and information cascade in networks has been extensively studied for decades in efforts to understand  the diffusion dynamics and its dependence on various factors, such as    network structures and infection rates. However, the inverse problem of inferring the source of diffusion based on limited observations is far less studied, and was first tackled by Shah and Zaman \cite{sz11} for identifying the source of rumor, where the rumor flows on edges according to independent exponentially distributed random times. A maximum likelihood (ML) estimator was proposed for maximizing the correct
localizing probability, and the notion of rumor-centrality was developed for approximately tracing back the source from the configuration of infected vertices at a given moment. The accuracy of estimations heavily depended on the structural properties of the networks. Shah and Zaman's model and their results on trees were   extended by Karamchandani and Franceschetti \cite{kf03}  to the case in which nodes reveal whether they have heard the rumor with independent probabilities. Along a different line, Pinto et al. \cite{ptv12} proposed  other ML estimators that perform  source detection via sparsely distributed observers who measure from which neighbors and at what time they received the information. The ML estimators were shown to be optimal for trees, and suboptimal for general networks under  the assumption that the propagation delays associated with edges are i.i.d. random variables with known Gaussian distribution.
 In contrast to previous probabilistic model for estimating the location of the source,    we study the problem from a combinatorial optimization's point of view; our goal is to find   an  observer set of minimum cost that guarantees   deterministic  determination of the accurate location of the source, i.e., to find a minimum weighted DRS.}

 The double resolvability is a strengthening of the  well-studied resolvability, where a vertex {$x$} resolves two vertices {$u,v$} if and only if {$d_G(u,x)\ne d_G(v,x)$.} A subset $S$ of $V$ is a {\em resolving set} (RS) of $G$ if every pair of vertices is resolved by some vertex of $S$. The minimum cardinality of a RS of $G$ is known as the {\em metric dimension} ${\tt md}(G)$ of $G$, which has been extensively  studied due to its theoretical importance and diverse applications (see e.g., \cite{cjmppsw07
cz03,elw12,hsv12} and references therein). Most literature on finding minimum resolving sets, {known as the {\em metric dimension problem},} considered the unweighted case.
The {unweighted} problem is $NP$-hard even for planar graphs,  split graphs, bipartite graphs and bounded degree graphs \cite{dpsv12,elw12,hsv12}. On general graphs, Hauptmann et al. \cite{hsv12} showed that the unweighted problem is not approximable within $(1-\varepsilon)\ln n$ for any $\varepsilon > 0$, unless $NP\subset
 DTIME(n^{\log \log n})$; moreover, the authors \cite{hsv12} gave a $(1+o(1))\ln n $-approximation algorithm based on approximability results of the test set  problem in bioinformatics \cite{bdk05}. A lot of research efforts have been devoted to {obtaining} the exact values or upper bounds of the metric {dimensions} of special graphic classes
 {\cite{cjmppsw07}}. Recently, Epstein et al \cite{elw12} studied the weighted version of the problem, and developed polynomial time exact algorithms for finding a minimum weighted  RS, when the underlying graph $G$ is  a cograph, a {\em $k$-edge-augmented tree} (a tree with additional $k$ edges) for constant $k\ge0$, or a (un)complete wheel.

Compared with nearly four decade research and vast {literatures} on   resolving sets (metric dimension), the study on DRS has a {relatively} short history and its results   have been very limited. The concept of DRS was introduced in 2007  by C\'aceres et al. \cite{cjmppsw07}, who proved that the minimum {RS} of the Cartesian product of graphs is tied in a strong sense to minimum DRS of the graphs:  the metric dimension of the Cartesian product of graphs $G_1$ and $G_2$ is upper bounded by   ${\tt md}(G_1)+{\tt dr}(G_2)$.
When restricted to the same graph, it is easy to see that a DRS must be a {RS}, but the reverse is not necessarily tru
. Thus ${\tt md}(G)\le{\tt dr}(G)$. The ratio ${\tt dr}(G)/{\tt md}(G)$ can be arbitrarily large. This can be seen from the tree graph $G$ depicted in Fig. \ref{fg:example}. On the one hand, it is easily checked that  $\{r_1,r_2\}$ is   a {RS} of $G$, giving  ${\tt dr}(G)\le 2$. On the other hand, ${\tt md}(G)=n/2$ since $\{s_1,s_2,\ldots,s_h\}$ is the unique minimum DRS of $G$, as proved later in Lemma \ref{tree} of this paper. In view of the large gap, algorithmic study on DRS deserves good
efforts, and it is interesting to explore the algorithmic relation between the minimum (weighted) DRS problem and \mbox{its resolving set counterpart.}

\begin{figure}[h]
\centerline{
  \includegraphics[width=6cm]{example.eps}}
  \caption{The graph tree $G$ with ${\tt dr}(G)=n/2$ and ${\tt md}(G)=2$.}\label{fg:example}
\end{figure}

Previous research on DRS considered only the unweighted case. As far as general graphs are concerned, the MDRS problem has been proved to be $NP$-hard \cite{kck09}, and solved experimentally by metaheuristic approaches that {use} binary encoding and standard genetic operators \cite{kck09} and that use variable neighborhood search \cite{mkkm12}.
To date, no efficient general-purpose algorithms with theoretically provable performance guarantees have been developed for the MDRS problem, let alone the MWDRS problem. Despite the NP-hardness, the approximability status of either problem has been unknown in literature. For special graphs, it is known that every RS of {Hamming graph}
is also a DRS \cite{kkcs12}. Recently, \v{C}angalovi\'{c} et al. showed that
${\tt dr}(G)\in\{3,4\}$ when $G$ {is a prism graph \cite{ckks13} or} belongs to one of two classes of convex polytopes \cite{kkcs2012}.

\iffalse
Diverse applications of resolving {sets} have been addressed in literature through years. However, reports on applications of DRS are few. It is worth pointing out that the MWDRS problem arises in locating the source of diffusion in a complex network. Because of the huge size of many real networks, it is often
impossible to observe the state of all vertices in a network.
It is desirable to find the location of the source (who initiates the diffusion) from measurements collected by sparsely placed {observers} \cite{ptv12}, where placing an observer at vertex $v$ incurs a cost $w(v)$, and the observer with a clock   can record the time at which the state of $v$ is changed (e.g.,
knowing a rumor, being infected or contaminated). There are different diffusion processes based on different assumption
. We consider the simultaneous spreading model: The time when the source originates an information is unknown. The information is diffused from the source to any vertex through shortest path
, where it takes one time unit for a vertex to send the information to all of its neighbors {simultaneously}.
It is typical that we {could only} observe the times when particular vertices get  the information but we {could} not observe who sent them \cite{glk10}.  The goal is to select a subset $S$ of vertices with minimum total cost $w(S)$ {such that}
the source can be uniquely located by placing observers on all vertices of $S$. This amounts to saying that $S$ is a {minimum weighted} DRS of $G$.  Indeed, a vertex subset $S'$  fails to locate the source if and only if there exist distinct $u,v\in V$ such that $S'$ cannot distinguish between the case of $u$ being the source and that of $v$ being the source, i.e., {$d_G(u,x)-d_G(u,y)= d_G(v,x)-d_G(v,y)$ for any $x,y\in S'$; equivalently, $S'$ is not a DRS of $G$.}
\fi

 \paragraph{Our contributions.} {As far as we know, our opening example of cost-effective source location is the first  real-world application of  {DRS} explicitly addressed. Motivated by the application, we study and provide a thorough treatment of the MWDRS problem in terms of algorithmic approximability. Broadly speaking, we}   show that the
 {MDRS and MWDRS  problems have similar approximability to their resolving set counterparts}.

 Based on the construction of Hauptmann et al.~\cite{hsv12}, we prove that there is an approximation preserving reduction from the minimum dominating set problem to the {MDRS} problem, showing that the MDRS problem does not admit $(1-\varepsilon)\ln n$-approximation algorithm for any $\varepsilon>0$ unless $NP\subset DTIME(n^{\log\log n})$. The strong inapproximability improves the $NP$-completeness established in \cite{kck09}. {Besides, we develop a $(\ln n+\ln\log_2n+1)$-approximation algorithm for solving the MWDRS problem in $O(n^4)$ time, based on a modified version of the approximation algorithms used in \cite{bdk05,hsv12}.}
 To the best of our knowledge, this paper is the first work providing explicit approximation lower and upper bounds for {the MDRS and MWDRS problems}, which are nearly tight {(for large $n$)}.  {A byproduct of our algorithm gives the first logarithmic approximation for the weighted metric dimension problem on general graphs.}

 {Despite many significant technical differences between handling DRS  and RS}, we establish the polynomial time solvability of the MWDRS problem for  all \bluecomment{these} graph classes, with one exception of cographs, where the {weighted metric dimension problem} is known to admit efficient exact algorithms \cite{elw12}. Our results are first known strong polynomial time algorithms for the MDRS problem  on  $k$-edge-augmented trees  and general wheels, including paths, trees and cycles. Using the fact that every minimum weighted DRS  is {\em minimal} (with respect to the inclusion relation), our algorithms make use of the graphic properties to cleverly ``enumerate'' minimal doubly resolving sets that are potentially minimum weighted, and select the best one among them.

 \medskip The paper is organized as follows: The inapproximability is proved in Section~\ref{sec:inapx}, The approximation algorithm for general graphs and exact algorithms for special graphs are presented in Sections \ref{sec:apx} and   \ref{sec:exact}, respectively. Future research directions are discussed in Section \ref{sec:conclude}. The omitted details are given in Appendix.

\section{Approximation lower bound}\label{sec:inapx}
 {In this section, we establish a logarithmic lower bound for approximation the MDRS problem under the assumption that $NP\not\subset DTIME(n^{\log\log n})$. {Hauptmann et al. \cite{hsv12}} constructed a reduction from the {\em minimum dominating set} (MDS) problem to the metric dimension   problem. {Although their proof does not work  for DRS,}   we show that their construction actually provides an approximation preserving reduction from the MDS problem to the MDRS problem.}

 A vertex subset $S$ of graph $G$ is a {\em dominating set} of $G$ if every vertex outside $S$ has a neighbor in $S$.   {The MDS problem is to find a dominating set of $G$ that has the minimum cardinality ${\tt ds}(G)$. Unless $NP\subset DTIME(n^{\log\log n})$, the MDS problem
 cannot be approximated within $(1-\varepsilon)\ln n$ for any $\varepsilon>0$ \cite{cc0
 }.}

\begin{lemma}\label{lem:np1}
There exists a polynomial time  transformation  that {transfers graph $G=(V,E)$
to graph $G'=(V',E')$
such that}   ${\tt dr}(G')\le {\tt ds}(G)+\lceil\log_2 n\rceil+3$.\qe
\end{lemma}

 {Let graphs $G$ and $G'$ be as in Lemma \ref{lem:np1}. It has been shown that, given any  RS (in particular DRS) $S$ of $G'$,  a dominating set of $G$ with cardinality at most $|S|$ can be found in polynomial time \cite{hsv12}.
{This, in combination with Lemma \ref{lem:np1}  and the logarithmic inapproximability} of the MDS problem \cite{cc0
}, gives the following lower bound for approximating minimum DRS.}
\begin{theorem}\label{th:lower}
Unless $NP\subset DTIME(n^{\log(\log n)})$, the {MDRS} problem cannot be approximated in polynomial time   within a factor of $(1-\epsilon)\ln n$, for any
 $\epsilon>0$.\qed
\end{theorem}
\section{Approximation algorithm}\label{sec:apx}
In this section, we {present} an $O(n^4)$ time approximation algorithm for the MWDRS problem in general graphs {that achieves} approximation ratio $(1+o(1))\ln n$, {nearly matching the lower bound $\ln n$ established in Theorem \ref{th:lower}.

Our algorithm {uses similar idea to that of Hauptmann et al.} \cite{hsv12} for approximating minimum resolving sets {in the metric dimension (MD) problem. The MD problem is a direct ``projection'' of the {\em unweighted} test set problem studied by Berman et al. \cite{bdk05} in the sense that a vertex in the MD problem can be seen as a ``test" in the test set problem, which allows Hauptmann et al. to apply Berman-DasGupta-Kao algorithm \cite{bdk05} directly. However, in the DRS problem, one cannot   simply} view two vertices as a ``test'', {because such a ``test'' would fail the algorithm in some situation.} {Besides,} the algorithm deals with only unweighted cases. Thus we {need  conduct certain transformation that transforms the DRS problem to a series of weighted test set problems. Furthermore, we need modify Berman-DasGupta-Kao algorithm to solve these {\em weighted} problems within logarithmic approximation ratios.}

\paragraph{Transformation.} For any $x\in V$, let $U_x=\{\{x,v\}:v\in V\setminus \{x\}\}$. {As seen later, each element of $U_x$ can be viewed as a {\em test} or a certain combination of {\em tests} in the test set  problem studied in \cite{bdk05}. From this point of view, we call each element of $U_x$ a {\em super test}, and consider the  {\em minimum weighted  super test set} (MWSTS) problem on $(V,U_x)$} as follows: For {each super} test $T=\{x,v\}\in U_x$, let its weight  be $w(T)=w(v)$, The problem is to find
  a set of super tests $\mathcal T\subseteq U_x$  such that each pair of vertices in $G$ is doubly resolved by some super test in $\mathcal T$ and the weight $w(\mathcal T)=\sum_{T\in\mathcal T}w(T)$ of $\mathcal T$ is minimized. {The following lemma establishes the relation between the MWDRS problem and the MWSTS problem}.

 \begin{lemma}\label{lem:star}
Let $S$ be a {DRS} of $G$ and $s\in S$. Then every pair of vertices in $G$ is {doubly} resolved by at least one element of $\{\{s,v\}:v\in S \setminus\{s\}\}$.
\end{lemma}
\begin{proof} Let $u,v$ be any two distinct vertices of $G$. There {exist} $s_1,s_2\in S $ {such that}   $d_G(u,s_1)-d_G(v,s_1)\ne d_G(u,s_2)-d_G(v,s_2)$. It follows that either $d_G(u,s_1)-d_G(v,s_1)\ne d_G(u,s)-d_G(v,s)$ or $ d_G(u,s)-d_G(v,s)\ne d_G(u,s_2)-d_G(v,s_2)$, saying that $u$ and $v$ are doubly resolved by either $\{s,s_1\}$ or $\{s,s_2\}$.
\end{proof}
Since $V$ is a  {DRS} of $G$, Lemma \ref{lem:star} implies that  {$U_x$ doubly resolves $V$}. More importantly, Lemma \ref{lem:star} provides the following immediate corollary {that is crucial to our algorithm design.}
\begin{corollary}\label{cor:bound}
Let $S^*$ be a {minimum weighted} DRS of $G$ and {$\alpha\in S^*$.  Then the} minimum weight of a solution to the MWSTS problem on $(V,U_{\alpha})$ is at most $w(S^*)-w(\alpha)$.\qed
\end{corollary}

\paragraph{Approximation.} {In order to solve the {MWSTS} problem, we adapt Berman-DasGupta-Kao algorithm \cite{bdk05} to augment a set $\mathcal T $ ($\subseteq U_x$)  of super tests to be a feasible solution step by step.} We define  {\em equivalence relation} $\equiv^{\mathcal T}$ on $V$   by: two vertices {$u,v\in V$} are equivalent under $\equiv^{\mathcal T}$  if and only if {$\{u,v\}$  is not doubly resolved by  any test of  $\mathcal T$.} Clearly, the number of equivalence classes is non-decreasing with the size of $\mathcal T$. Let
  $E_1,\ldots,E_k$ be the equivalence classes of $\equiv^{\mathcal T}$.  The value
    $H_{\mathcal T}:=\log_2(\prod_{i=1}^{k}|E_i|!)$ is called the {\em entropy} of $\mathcal T$.
    {Note that
    \begin{eqnarray}\label{eqv}
    \begin{array}{rcl}
    H_{\mathcal T}=0&\Leftrightarrow&\text{every equivalent class of $\equiv^{\mathcal T}$ is a singleton}\\
    &\Leftrightarrow&\cup_{T\in \mathcal T}T\text{ is a DRS of $G$.}
    \end{array}
    \end{eqnarray}
     Hence our task is reduced to finding a set $\mathcal T$ of super tests with zero entropy $H_{\mathcal T}$ and weight $w(\mathcal T)$ as small as possible.}

    {For any super test   $T\in U_x $},
   an equivalence class   of $\equiv^{\mathcal T}$ is either an equivalence class of $\equiv^{\mathcal T\cup T}$ or it is partitioned into several {(possibly more than two)} equivalence classes of $\equiv^{\mathcal T\cup T}$. {(If $T$ partitions each equivalent class into at most two equivalent classes, then $T$  works as a test in the test set problem.)} Therefore $H_{\mathcal T}\ge H_{\mathcal T\cup T}$, and $IC(T,\mathcal T):=H_{\mathcal T}-H_{\mathcal T\cup T}\ge0$
     equals  the decreasing amount of the entropy when adding $T$ to $\mathcal T$. It is clear that
     \begin{gather}\label{bound}
     IC(T,\emptyset)\le \log_2n!-log_21< n\log_2n.\end{gather}

We  {now give a $(1+o(1))\ln n$-approximation algorithm  for the {MWSTS} problem  on $(V,U_x)$}.  The algorithm adopts the   greedy heuristic  to decrease the entropy of the current set of super tests  \mbox{at a minimum cost (weight).}

\medskip
\hrule
\vspace{-1mm}\begin{algorithm}\label{alg1}
Finding minimum weighted set $\mathcal T$ of super sets.
\vspace{1mm}\hrule
{\small
\begin{enumerate}
  \vspace{-0mm}\item \hspace{1mm}  {$\mathcal T\leftarrow\emptyset$}
  \item \hspace{1mm} \textbf{while} $H_{\mathcal T}\neq 0$ \textbf{do}
  \item \hspace{5mm} Select a {super test} $T\in U_x-\mathcal T$ that \emph{maximizes} $\frac{IC(T,\mathcal T)}{w(T)}$
  \item \hspace{5mm} {$\mathcal T\leftarrow\mathcal T\cup T$}
  \item \hspace{1mm} \textbf{end-while}
\end{enumerate}
}
\vspace{-0mm}\hrule
\end{algorithm}

The major difference between Algorithm \ref{alg1} and the algorithms in \cite{bdk05,hsv12} is the criterion used in Step 3 for selecting $T$. It generalizes the previous   unweighted setting.
The following lemma
extends the result
on test set \cite{bdk05} to super test set.
\begin{lemma}\label{ich}
  $IC(T,\mathcal T_0)\ge IC(T,\mathcal T_1)$ for any sets $\mathcal T_0$ and $\mathcal T_1$ of super tests with \mbox{$\mathcal T_0\subseteq\mathcal T_1$.}\qed
\end{lemma}

Using (\ref{bound}) and Lemma \ref{ich}, the proof of performance ratio goes almost verbatim as the argument of Berman et al. \cite{bdk05}. We include a proof in Appendix \ref{apx1} for completeness.

\begin{theorem}\label{th:performance}
Algorithm \ref{alg1} is an $O(n^3)$ time   algorithm for {the MWSTS   problem} on $(V,U_x)$ with {approximation} ratio {\mbox{$\displaystyle\ln\left( \max_{T\in U_x}IC(T,\emptyset)\right)\!+\!1\!\le\!\ln n\!+\!\ln\log_2 n\!+\!1$.$\Box$}}
\end{theorem}

Suppose that given the MWSTS   problem on $(V,U_x)$, Algorithm \ref{alg1}  outputs a super test set $\mathcal T_x$. By (\ref{eqv}),
  Running Algorithm \ref{alg1} for $n$ times, we obtain $n$ {doubly resolving  sets} $S_x$, $x\in V$ of $G$, from which we select the one, say $S_v$, that has the minimum weight,
i.e. $w(S_v)=\min\{w(S_x):x\in V\}$.
\begin{theorem}
The {MWDRS} problem can be approximated in $O(n^4)$ time within a ratio \[\displaystyle\ln\left( \max_{u,v\in V}IC(\{u,v\},\emptyset)\right)+1\le \ln n+\ln\log_2n+1=(1+o(1))\ln n.\]
\end{theorem}
\begin{proof}
Let $S^*$ be an optimal solution to the MWDRS problem. It suffices to show $w(S_v)/w(S^*)\le (1+o(1))\ln n$. Take $\alpha\in S^*$, and let $\mathcal T^*_{\alpha}$ be an optimal solution to the MWSTS problem on $(V,U_{\alpha})$. It follows from {the} choice of $S_v$, Theorem \ref{th:performance} and Corollary \ref{cor:bound} that $w(S_v)\le w(S_{\alpha})=w(\alpha)+w(\mathcal T_{\alpha})\le w(\alpha)+(\ln n+\ln\log_2n+1)  w(\mathcal T_{\alpha}^*)<(\ln n+\ln\log_2n+1) w(S^*)$.
\end{proof}

Our algorithm and analysis show that the {algorithm}
of   \cite{bdk05} can be extended to solve the weighted test set problem, where each test has a nonnegative weight, by changing the selection criterion to be maximizing $IC(T,\mathcal T)$ divided by the weight of $T$. A similar extension applied to the algorithm of Hauptmann et al.~\cite{hsv12} gives a $(1+o(1))\ln n$-approximate solution to the {weighted metric dimension} problem.

\section{Exact algorithms}\label{sec:exact}
{Let $k\ge0$ be a constant. A connected graph is called a {{\em $k$-edge-augmented tree}} if the removal of  {at most $k$ edges} from the graph leaves a spanning tree. Trees and cycles are 0-edge- and {1-edge-augmented} trees, respectively. We design efficient algorithms for solving the MWDRS problem {exactly} on $k$-edge-augmented trees.}
 Our algorithms run in linear time for $k=0,1$, and
in $O(n^{12k})$ time for $k\ge2$.

{A graph is called a general wheel if it is formed from a cycle by adding a vertex and joining it to some (not necessarily all) vertices on the cycle. We solve the MWDRS problem
on general wheels   in cubic time by dynamic programming.}

\subsection{$k$-edge-augmented trees} Let $G=(V,E)$ be a $k$-edge-augmented tree, and let $L$  be the set of leaves (degree 1 vertices) in $G$. For simplicity, we often use $d(u,v)$ instead of $d_G(u,v)$ to denote the distance between vertices $u,v\in V$ in the underlying graph $G$ of the MWDRS problem.

\subsubsection{Trees: the case of $k=0$.}
When $k=0$, graph $G=(V,E)$ is a tree.  {There is a fundamental difference between DRS and RS of $G$ in terms of minimal sets. In general, $G$ may have multiple minimal RSs and even multiple minimum weighted  RSs. Nevertheless, in any case $G$}   has only one {minimal} DRS, {which consists of all its leaves}.  In particular, we have ${\tt dr}(G)=|L|$.
\begin{lemma}\label{tree}
$L$ is the unique minimal DRS of $G$.
\end{lemma}
\begin{proof}
For any two vertices $u,v\in V$, there exist leaves $l_1,l_2\in L$ such that  the path between $l_1$ and $l_2$ goes through $u$ and $v$. It is easy to see that
$d(u,l_1)-d(u,l_2)\neq d(v,l_1)-d(v,l_2)$. So $L$ is a DRS. On the other hand, consider any leaf $l\in L$ and its neighbor $p\in V$. Since $d(l,v)-d(p,v)=1$ for any $v\in V-\{l\}$, we see that each DRS of $G$ contains $l$, and thus $L$. The conclusion follows.
\end{proof}

\subsubsection{Cycles: a special case of  $k=1$.}\label{cyc}
Let $G=v_1v_2\cdots v_nv_1$ be a cycle, where  $V=\{v_1,v_2,\ldots,v_n\}$. Suppose without loss of generality that $w(v_p)=\min_{i=1}^nw(v_i)$, {where $p:=\lceil n/2\rceil$.}
{It was known  that any pair of vertices whose distance is not exactly $n/2$ is a minimal RS of $G$, and vice versa \cite{elw12}.}   {As the next lemma shows, the characterization of   DRS turns out to be more complex.}
{Each nonempty subset  $S$ of $V$ cuts $G$ into}
a set $\mathcal P_S$ of edge-disjoint paths such that they are internally disjoint from $S$ and their union is $G$.

\begin{lemma}\label{n/2}
{Given a cycle $G=(V,E)$, let $S$ be a {nonempty} subset of  $V$.  Then $S$ is a DRS of $G$ if and only if no path in $\mathcal P_S$ has length longer than $\lceil n/2\rceil$  and at least one path in $\mathcal P_S$ has length shorter than $n/2$.\qed}
\end{lemma}
An instant corollary  reads:
The size of a minimal DRS of cycle $G$ is $2$ or $3$ when $n$ is odd, and is $3$ when $n$ is even; In particular, ${\tt dr}(G)=2$ when $n$ is odd, and ${\tt dr}(G)=3$ when $n$ is even. {These properties together with the next one lead to our algorithm for solving the MWDRS problem on cycles.}
\begin{corollary}\label{cor:vp}
{If some minimum weighted DRS has cardinality $3$, then there exists a minimum weighted DRS  of $G$ that contains vertex $v_p$.\qed}
\end{corollary

\medskip
\hrule
\vspace{-1.5mm}\begin{algorithm}\label{alg2} {Finding minimum weighted DRS $S$ in cycle $G$.}
\vspace{1mm}\hrule
{\small
\begin{enumerate}
\vspace{-1mm}  \item $\omega\leftarrow w(v_1)$, $i[1]\leftarrow 1$, $j\leftarrow 1$, $W\leftarrow w(V)$
  \item {\bf for} $h=1$ {\bf to} $p$ {\bf do}
  \item \hspace{3mm}{\bf if} $w(v_h)<\omega$ {\bf then} $j\leftarrow j+1$, $i[j]\leftarrow h$, $\omega\leftarrow w(v_h)$
  \item {\bf end-for}
\item {\bf if} $j>1$ {\bf then} $k\leftarrow j$ {\bf  else} $k\leftarrow2$, $i[k]\leftarrow p$
\item {\bf if} $n$ is odd  {\bf then} $S\leftarrow \arg\min_{i=1}^nw(\{v_i,v_{i+p-1}\})$, $W\leftarrow w(S)$
\item {\bf for} $j=1$ {{\bf to}} $k-1$ {\bf do}
\item \hspace{3mm} {let $u_j$ be a vertex in $ \{v_h: i[j]+p\!\le\! h\!\le\! i[j+1]+p\}$ with \mbox{$w(u_j)=\min_{h=i[j]+p}^{i[j+1]+p}w(v_h)$}}
\item \hspace{3mm} {\bf if} $w(v_p)+w(v_{i[j]})+w(u_j) <W$ {\bf then} $S\leftarrow \{v_p,v_{i[j]},u_j\}$, {$W\leftarrow w(S)$}
\item {\bf end-for}
\end{enumerate}
}
\vspace{-1.5mm}\hrule
\end{algorithm}
Note that $V_j:=\{v_h: i[j]+p\le h\le i[j+1]+p\}$, $j=1,\ldots,k-1$ induce $k-1$  internally disjoint paths in $G$. {It is thus clear that}  Algorithm \ref{alg2} runs in $O(n)$ time. The vertices indices  $1=i[1]< i[2]<\cdots< i[k]=p$ found by the algorithm satisfy  $w(v_{i[j]})=\min_{h=i[j]}^{i[j+1]-1}w(v_h)=\min_{h=1}^{i[j+1]-1}w(v_h)$ for every $j=1,\ldots,k-1$ and $w(v_{i[k]})=w(v_p)=\min_{h=1}^{p}w(v_h)$. Moreover, either $w(v_1)=w(v_p)$ and $k=2$, or $w(v_{i[j]})> w(v_{i[j+1]})$ for every $j=1,\ldots,k-1$. {These facts together with the properties mentioned above verify the correctness of the algorithm.}

\begin{theorem}\label{th:cycle}
{Algorithm \ref{alg2} finds in  $O(n)$  time a minimum weighted DRS of cycle $G$.}\qed
\end{theorem}

\subsubsection{The case of general $k$.}
{Our approach resembles  at a high level the one used by Epstein et al. \cite{elw12}.} {However, double resolvablity imposes more strict restrictions, and  requires extra care to overcome technical difficulties.}
Let $G_b=(V_b,E_b)$ be the graph obtained from $G=(V,E)$ by repeatedly deleting leaves.  {We call $G_b$ the {\em base graph} of $G$.} We reduce the MWDRS problem on $G$ to the MWDRS problem {on} $G_b$ (see Lemma \ref{lem:base}). The latter problem can be solved in polynomial time by exhaustive enumeration, {since, \bluecomment{as proved   in the sequel}, every minimal DRS of $G_b$ has cardinality at most $12(k-1)$ for $k\ge 2$}

 Clearly, $G_b$ is connected and has minimum degree at least 2. A vertex in $V_b$ is called a {\em root} if in $G$ it is adjacent to some vertex in $V\setminus V_b$. Let $R$ denote the set of  roots. Clearly, $R\cap L=\emptyset$.  In $G_b$, we change the weights of all roots to zero, while the weights of other vertices remain the same as in $G$.
\begin{lemma}\label{lem:base}
Suppose that $S_b$ is a minimum weighted DRS of $G_b$. Then $(S_b\setminus R)\cup L$ is a minimum weighted DRS of $G$.\qed
\end{lemma}
 {Therefore,} for solving the MWDRS problem on a weighted $k$-edge-augmented tree $G$, we only need to find a minimum weighted DRS of base graph $G_b$ with the weights of all roots modified to be $0$.

 For 1-edge-augmented tree $G$, its base graph $G_b$ is a cycle,   whose minimum weighted DRS can be found in $O(n)$ time  {(recall Algorithm \ref{alg2})}. Combining this with  {Lemmas \ref{tree} and \ref{lem:base},} we have the following {linear time solvability}.
\begin{theorem}
There is an $O(n)$ time exact algorithm for solving the MWDRS problem on $k$-edge-augmented trees, for $k=0,1$, {including trees and cycles.}
\end{theorem}

In the remaining discussion for $k$-edge-augmented tree, we assume $k\ge 2$. A vertex is called a {\em branching vertex} of a graph if it has degree at least 3 in the graph. Recall that every vertex of
the base graph  $G_b=(V_b,E_b)$ has degree  at least $2$.
{It can be shown that (see Lemmas \ref{lem:path} and \ref{lem:four} in Appendix \ref{apx2})
  \begin{itemize}
  \item In $O(|E_b|)=O(n^2)$ time, $G_b$ can be decomposed into at most $ 3k-3$ {edge-disjoint paths whose ends are branching vertices of $G_b$ and internal vertices have degree 2 in $G_b$.
\item For any minimal DRS set $S$ of $G_b$, and any path $P$ in the above path decomposition of $G_b$,    at most four vertices of $S$ are contained in $P$.
\end{itemize}
It follows that} every minimal DRS of $G_b$ contains at most $12(k-1)$ vertices. Our algorithm for finding the minimum weighted DRS of $G_b$ {examines}   all {possible}  subsets   of $V_b$ with cardinality at most $12(k-1)$ {by taking at most four vertices from each path in the path decomposition of $G_b$}; among these sets, the algorithm selects a DRS of $G_b$ {with} minimum weight. Have a table that stores the distances between each pair of vertices in $G_b$, {it takes $O(n^2)$ time to  test the double resolvability of a set.} Recalling Lemma \ref{lem:base}, we obtain the following strong polynomial time solvability for the general $k$-edge-augmented trees.
\begin{theorem}\label{th:ktree}
The {MWDRS} problem on $k$-edge-augmented trees can be solved in $O(n^{12k})$ time. \qed
\end{theorem}

\subsection{Wheels}
A  general {\em wheel}  $G=(V,E)$ on $n$ \bluecomment{$(\ge6)$} vertices  $v_1,v_2,\ldots,v_n$  is formed by the {\em hub} vertex $v_n$ and a cycle {$C=(V_c,E_c)$} over the vertices  {$v_1,v_2,\ldots,v_{n-1}$}, called {\em rim} vertices, {where the hub is adjacent to some (not necessarily all) rim vertices}. {We develop dynamic programming algorithm to solve the MWDRS problem on general wheels in $O(n^3)$ time.}

 We start  with complete wheels whose DRS has a very nice characterization that \bluecomment{is} related to the consecutive one property. A general wheel is {\em complete} if its hub is adjacent to every rim vertex.
\begin{equation}\label{1or2}
\text{{The distance in $G$ between any two vertices in $G$ is} either $1$ or $2$.}
 \end{equation
\begin{lemma}\label{cw}
Given a complete wheel $G=(V,E)$, let $S$ be a proper nonempty subset of $V$. Then $S$ is a doubly resolving set  if and only if $S\cap V(C)$ is a  dominating set of $C$ and   any pair of rim  vertices outside $S$ has at least two neighbors in {$S\cap V(C)$}.
\end{lemma}
\begin{proof}
If $S\cap V(C)$ is not a   dominating set of $C$, then there exists a vertex   $v_i \in V(C)- S$  such that $v_i$ is not adjacent to any vertex of $S\cap V(C)$. In  this case, $S$ cannot doubly resolve $\{v_n,v_i\}$ because for any $s_1,s_2\in S$, $d(s_1,v_n)-d(s_1,v_i)=-1=d(s_2,v_n)-d(s_2,v_i)$.

If $S\cap V(C)$ is a  dominating set of $C$ but there exist two cycle vertices  {$v_i,v_j \in V(C)-S$ such that $v_i,v_j$} are uniquely dominated by the same cycle vertex $v\in S$, then for any two vertices $s_1,s_2\in S$, $d(s_1,v_i)-d(s_1,v_j)=0=d(s_2,v_i)-d(s_2,v_j)$, saying that $S$ is not a doubly resolving set.

 Suppose that $S$ satisfies the condition stated in the lemma. We prove that $S$ can resolve every pair of vertices $x,y$ in $G$. When one of $x$ and $y$, say $x$, is a rim vertex in $S$,
since $n\ge 6$, there exists  another rim vertex $z\in S-\{x\}$
that is not adjacent to  $x$. It follows from (\ref{1or2}) that $\{x,y\}$ is resolved by $\{x,z\}$ as $d(x,x)-d(x,z)=-2<-1\le d(y,x)-d(y,z)$. When both $x$ and $y$ are rim vertices outside $S$, there are two rim vertices $x'$ and $y'$ in $ S$ dominating $x$ and $y$, respectively. It follows that $\{x,y\}$ is resolved by $\{x',y'\}$. When one of $x$ and $y$, say $x$ is the hub, we only need consider the case of $y$ is a rim vertex outside $S$. Take rim vertices $z,z'$ from $S$ such that $z$ dominates $y$ and $z'$ does not dominate $y$. It follows that $\{x,y\}$ is resolved by $\{z,z'\}$ as $d(x,z)-d(x,z')=0<-1=d(y,z)-d(y,z')$.
\end{proof}

{The characterization in Lemma \ref{cw} can be rephrased as follows: A subset $S\subseteq V$ is a DRS of $G$ if and only if every set of three consecutive vertices on $C$ contains at least one vertex of $S$, and every set of five consecutive vertices on $C$ contains at least two vertices of $S$. This enables us to formulate the MWDRS problem on a complete wheel as an integer programming with {consecutive 1's and circular 1's constraints}, which can be solved in {$O(n^3\log^2n)$} time by Hochbaum and Levin's algorithm \cite{hl06}.} {(To the best of our knowledge, there is no such a concise way to formulate the metric dimension problem on complete wheels  as an integer programming with consecutive one matrix.)} {Moreover, it is not hard to see from the characterization that linear time efficiency can be achieved by dynamic programming approach. Furthermore, we elaborate on the idea to solve the MWDRS problem on more complex general wheels.}

\begin{theorem}\label{general}
The {MWDRS} problem on complete wheels can be solved in $O(n)$ time. The {MWDRS} problem on general wheels can be solved in  $O(n^3)$ time.\qed
\end{theorem}
\section{Conclusion}\label{sec:conclude}
{In this paper, we have established  $\Theta(\ln n)$ approximability of the {MDRS and MWDRS problems} on general graphs. There is still a gap of {$1+\ln\log_2n$} hidden in the big theta (see Theorems \ref{th:lower} and \ref{th:performance}). It deserves good research efforts to obtain even tighter upper bounds for the approximability. The $k$-edge-augmented trees, general wheels and cographs are known graph classes on which the {weighted metric dimension  problem} is polynomial time solvable. In this paper, we have extended the polynomial time solvability to the {MWDRS} problem for the first two graph classes. It would be interesting to see whether the problem on cographs {and} other graphs also admits efficient algorithms.}

redpajama_set_name: RedPajamaArXiv


Ella to Discuss Privacy, Data for Business Lawyers at CLE - Trepanier MacGillis Battina P.A.
September 29, 2016 by Trepanier MacGillis Battina P.A.
Trepanier MacGillis Battina P.A. shareholder V. John Ella will be discussing privacy law at "Data for Business Lawyers: Analytics & Sharing; Privacy & Security; Breach Response; Mobil Device Security" presented by Minnesota CLE on Monday, October 24, 2016 from 9:00 am to 12:15 pm. Webcast attendance is available.
V. John Ella is a privacy and defamation attorney.


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text:
Sankt Urban hotels
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Best Sankt Urban hotels
Erlebnishaus Spiess
Set in Feldkirchen in Kärnten, in a building dating from 1972, Erlebnishaus Spiess offers a garden and guestrooms with free WiFi. Featuring luggage storage space, this property also provides guests with a children's playground. The units at the guest house come with a seating area, a flat-screen TV with satellite channels and a private bathroom with a hairdryer and a shower. All guest rooms feature a wardrobe. Erlebnishaus Spiess offers a continental or buffet breakfast. The accommodation offers a terrace. You can play table tennis at Erlebnishaus Spiess, and the area is popular for hiking. Bad Kleinkirchheim is 34 km from the guest house, while Villach is 37 km away. The nearest airport is Klagenfurt Airport, 22 km from Erlebnishaus Spiess.
Haus Zur Sonne
Set in Agsdorf in the Carinthia Region, 29 km from Bad Kleinkirchheim, the home stay Haus zur Sonne boasts a barbecue and terrace. Free private parking is available on site. Each room is equipped with a flat-screen TV with satellite channels. Certain rooms include views of the mountain or garden. The rooms are equipped with a private bathroom. Haus zur Sonne features free WiFi and singing bowl massages are offered. You can play table tennis at the homestay. The homestay also offers free use of bicycles. Villach is 28 km from Haus zur Sonne, while Bled is 41 km from the property. The nearest airport is Klagenfurt Airport, 15 km from Haus zur Sonne.
Hotel Urbani
When you stay at HOTEL Urbani in Steindorf am Ossiacher See, you'll be on a lake, within a 15-minute drive of Landskron Castle and Gerlitzen Ski Resort. Featured amenities include laundry facilities and an elevator (lift). Free self parking is available onsite.
Gasthof Ogertschnig
Located in Klagenfurt and with Pitzelstätten Castle reachable within 1.3 km, Gasthof Ogertschnig provides a tour desk, non-smoking rooms, a garden, free WiFi and a bar. Featuring an ATM, this property also welcomes guests with a restaurant and a sun terrace. Guests can enjoy river views. All units at the guest house are equipped with a seating area, a flat-screen TV with satellite channels and a private bathroom with a hairdryer and a shower. The rooms include a desk. A continental breakfast is available daily at Gasthof Ogertschnig. Cycling is among the activities that guests can enjoy near the accommodation. Ehrenbichl Castle is 2.5 km from Gasthof Ogertschnig, while Drasing Castle is 3 km away. The nearest airport is Klagenfurt Airport, 8 km from the guest house.
Ferien am Talhof
Ferien am Talhof is located 3 km from the centre of Sankt Urban, surrounded by scenic nature and right at the shores of a swimming lake. The bright apartments at the Talhof have a balcony, a sofa bed and satellite TV. The fully equipped kitchen or kitchenette has a refrigerator and a dining area. Bed linen and towels are included in the price and guests can make use of the laundry room. The barbecue facilities on the sunny terrace are at guests' disposal. A children's playground can be found in the garden and activities for children are organized. A billiards table and a ping-pong table are also provided. Ferien am Talhof offers horseback riding and free fishing in the bathing lake. Guests can rent free go-karts in nice weather and bobsleighs in winter. Simonhöhe ski area can be reached within 4.4 km.
Seepark Hotel - Congress & Spa
Located in Klagenfurt am Woerthersee, Seepark Hotel - Congress & Spa is on a lake, a 3-minute walk from University of Klagenfurt and 7 minutes by foot from Minimundus. Featured amenities include a business center, complimentary newspapers in the lobby, and dry cleaning/laundry services. Planning an event in Klagenfurt am Woerthersee? This hotel has facilities measuring 350 square feet (33 square meters), including a conference center. Free self parking is available onsite.
Landgasthof Ploschenberg
Located in Köttmannsdorf in the Carinthia Region, Landgasthof Plöschenberg features a children's playground and sun terrace. Guests can enjoy the on-site restaurant. Free private parking is available on site. The rooms come with a flat-screen TV with satellite channels. The rooms include a private bathroom. Extras include free toiletries and a hairdryer. Landgasthof Ploeschenberg features free WiFi throughout the property. Villach is 30 km from Landgasthof Plöschenberg, while Bled is 25 km from the property. Klagenfurt Airport is 10 km away.
Landidyll Hotel Nudelbacher
Located in Feldkirchen in Kaernten, Landidyll Hotel Nudelbacher is in a rural location, within a 10-minute drive of Maltschacher See and Ossiacher Lake. Featured amenities include complimentary newspapers in the lobby, luggage storage, and a safe deposit box at the front desk. Free self parking is available onsite.
Hotel Palais Porcia
With a stay at Hotel Palais Porcia, you'll be centrally located in Klagenfurt am Woerthersee, steps from Dragon Fountain and within a 5-minute walk of Old Square. Featured amenities include a computer station, express check-in, and complimentary newspapers in the lobby. This hotel has 2 meeting rooms available for events. Self parking (subject to charges) is available onsite.
Pros: Great value,reasonably priced, Super location, Friendly and accommodating staff!
Cons: Brandy at the bar
Hotel Plattenwirt
When you stay at Hotel Plattenwirt in Klagenfurt am Woerthersee, you'll be on a lake, within a 15-minute walk of Minimundus and Worther Lake. Featured amenities include a computer station, complimentary newspapers in the lobby, and dry cleaning/laundry services. Free self parking is available onsite.
City Hotel Zum Domplatz
With a stay at City Hotel Zum Domplatz, you'll be centrally located in Klagenfurt am Woerthersee, just a 4-minute walk from New Square and 6 minutes by foot from Landhaus. Featured amenities include a computer station, express check-in, and complimentary newspapers in the lobby.
Hotel Weidenhof
Located in the outskirts of Klagenfurt, Hotel Weidenhof is 700 metres from the southern shores of Lake Wörth away. It offers free parking and a breakfast buffet. All spacious rooms provide a bathroom, a TV and a toilet. Some also have a seating area and a balcony. Guests of the Weidenhof Hotel can enjoy Austrian and international cuisine in the restaurant or outside on the terrace. The beautiful garden invites to relax in the shade or to take a sunbath.
This lovely hotel is situated in St Urban. Pets are not permitted on the premises.
Hotel Sandwirth
A stay at Hotel Sandwirth places you in the heart of Klagenfurt am Woerthersee, a 2-minute drive from Old Square and 5 minutes from Happ's Reptilienzoo. Featured amenities include complimentary wired Internet access, a 24-hour business center, and complimentary newspapers in the lobby. Planning an event in Klagenfurt am Woerthersee? This hotel has facilities measuring 2153 square feet (200 square meters), including a conference center. Self parking (subject to charges) is available onsite.
Pension Fischinger
Pension Fischinger is located 3,500 metres away from the centre of Feldkirchen and 4 km froom Maltschach lake. The on-site indoor pool, which is open from 27 April till October, the sauna and the steam bath can be used free of charge. All rooms come with a private bathroom and a flat-screen TV with satellite and radio channels. Free WiFi is available throughout the building and the property is decorated with pictures by a local artist Guests of Fischinger's Pension can enjoy a breakfast every morning, and can have dinner at the restaurant on site. Several swimming lakes can be found within a maximum range of 15 km. The Gerlitzen Ski Area is 15 km and Bad Kleinkirchheim is 45 km away. Due to it's central location, the property is an ideal starting point for visits to various sites and landmarks in the Carinthia region.
Carinthia Stadthotel
This 4-star hotel is located in the centre of Klagenfurt, just a short walk from the cathedral and all major sights. It offers free in-room WiFi. The air-conditioned rooms at Carinthia Stadthotel are traditionally furnished and feature a flat-screen cable TV, a minibar, and a bathroom with hairdryer. A rich breakfast buffet is served every morning. Hotel Carinthia also features a bar. A limited number of private parking spaces is available on site free of charge. Public short-term parking zones are available in the surrounding area, which can be used free of charge between 18:00 and 08:00 and from 12:00 on Saturday to 08:00 on Monday. Reduced tickets for the Domgarage (a public parking garage a 5-minute walk away) are available at the reception.
Gasthof Waldwirt
Frühstückspension Kölich
Featuring free WiFi throughout the property, Frühstückspension Kölich offers pet-friendly accommodation in Klagenfurt, 6 km from the Strandbad Klagenfurt Public Bathing Beach. Free private parking is available on site. Some units have views of the garden or city. A flat-screen TV with satellite channels is provided. There is a shared kitchen at the property. The Minimundus miniature park is 5.1 km from the property. The nearest airport is Klagenfurt Airport, 6 km from Frühstückspension Kölich.
Jugend- Und Familiengästehaus Klagenfurt
Just 500 metres from Alpen-Adria-Universität Klagenfurt, the Jugend- und Familiengästehaus Klagenfurt offers non-smoking accommodation with a lounge featuring free Wi-Fi. All private and dormitory rooms come with a bathroom located in the unit. Some rooms also have a balcony, while double rooms additionally offer cable TV. Rooms with disability access are available on the ground floor, upon request and availability. Breakfast is served each morning at the Jugend- und Familiengästehaus. The city centre, with shops and restaurants can be reached in a 5-minute bus ride. The ATK Klagenfurt Jugendgästehaus Bus Stop is 100 metres from the property. Europapark and Minimundus Park are a 10-minute walk away and the Wörthersee Lake is 1.8 km away.
Hotel Goldener Brunnen
The hotel lies in the old town of Klagenfurt overlooking the Wörthersee lake. Klagenfurt's main station can be reached in 12 minutes on foot. The hotel is within minutes of numerous bars, restaurants and nightspots. Klagenfurt Airport is less than a 15-minute drive away. This modern 26-room hotel is perfect for both families and business trips. There are rooms with connecting doors available and each room has a minibar. The hotel features a romantic courtyard with a café that serves traditional Austrian dishes. Wireless Internet access and parking are available. In addition to a lobby area with lift access to the upper floors, further facilities available to guests at this business hotel include a conference room and bike hire service. In addition to a private bathroom with a hairdryer, in-room amenities include a direct dial telephone, satellite/cable TV, radio and Internet access, as well as a minibar and tea and coffee making facilities. A safety deposit box is a further standard feature of all accommodation units. Fans of the fairway can head to the nearest golf course by the lake, which is approximately 45 minutes' walk from the hotel. A buffet breakfast is served each morning at the hotel.
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text:
Patty Jackson: The 4-1-1
My Nephew's Back; Omarosa's Back; Aretha's Chosen
posted by Patty Jackson - Feb 6, 2018
WDAS presents our Fourth Annual Women Of Excellence Luncheon Saturday, March 31st. This year's honorees include Broadcast Legend Rev. Louise Williams Bishop, City Councilwoman Cherelle Parker, TV Icon Lisa Thomas Laury, First Lady of Mother Bethel AME Church Leslie Patterson-Tyler, The Philadelphia Tribune's own Shonda McClain and our Special Trailblazing Woman of Excellence honor to Hip Hop Legend and Oscar Nominee Queen Latifah! It all takes place Saturday, March 31st from 11am-4pm at The Sheraton Philadelphia Downtown, 17th and Race. All the details are at wdasfm.com.
Jennifer Hudson will play Aretha! Clive Davis confirmed at his annual Grammy Party that The Queen Of Soul, Aretha Franklin, officially gave the nod to Oscar Winner and American Idol Alumnus Hudson. No details yet on when the bio will begin filming, but Hudson's selection is a big step forward to getting Franklin's incredible life story to the big screen.
The Simone Biles Story: Courage To Soar is coming to Lifetime. Based on the book written by the 19-time Olympic and World Championship medal winner, the TV drama will focus on Biles' journey and her parents support throughout winning gold at the 2016 Olympic Games. Tisha Campbell-Martin (My Wife & Kids; Martin) and Julius Tennon (How To Get Away With Murder) star as Biles' parents and Jeante' Godlock stars as Simone. The Simone Biles Story: Courage To Soar premieres on Lifetime this Saturday, February 3rd at 8pm.
Speaking of Julius Tennon from How To Get Away With Murder, that Viola Davis drama will do a crossover episode with Kerry Washington's Scandal. Details are being kept close, but each leading lady took instagram shots on the other's set: Washington in the courthouse and Davis in the White House. Both shows are produced by Shonda Rhimes as part of the blockbuster ABC-TGIT lineup. Scandal, which is in its final season, has welcomed Hillary Clinton, Oprah Winfrey and Kobe Bryant to the set. Scandal will also air Kerry Washington's directed episode this season. The Scandal and How To Get Away With Murder crossover event airs Thursday, March 1st beginning at 9pm on ABC after Grey's Anatomy.
Fly Eagles Fly. The Philadelphia Eagles are returning to the Superbowl. The NFC champions will take on the New England Patriots this Sunday. Broadway Star, Tony Award Winner and Philly's own Leslie Odom, Jr. will sing America The Beautiful at the game. Also, Justin Timberlake is back for the halftime show, and Doylestown's own Pink will do the National Anthem. The Superbowl airs on NBC this Sunday, February 4th.
Has Omarosa Manigault-Newman landed on CBS' Celebrity Big Brother? Yes. Reality's favorite TV villainess is returning to her roots. Omarosa's housemates will included Keshia Knight Pulliam (The Cosby Show; House Of Payne), Ross Matthews (The Tonight Show; E! Entertainment), TV Personality/Singer Mark McGrath of Sugar Ray and Brandi Glanville (Real Housewives Of Beverly Hills). The former member of Donald Trump's administration also signed with a booking agent and was thought to be doing $50,000 a shot speaking engagements. Could Big Brother impact that and the story she said America wanted to hear but earned a Bye Felicia from Robin Roberts? Celebrity Big Brother premieres on CBS Wednesday, February 7th at 8pm.
ESPN's Jemele Hill has been ousted from her co-anchor spot on the weekday edition of SportsCenter. ESPN has cut talent and staff recently, but many are wondering if something else is involved here. Hill was suspended last year by ESPN after she used her personal twitter to call President Donald Trump a white supremacist. White House Spokesperson Sara Huckabee Sanders said such tweets were a fire-able offense. Hill apologized for the fact that ESPN was now involved, but not for the tweet. Hill also suggested boycotting sponsors was an option for those who didn't like team owners pressuring their players to stand for the National Anthem. With three years left on her four year ESPN contract, the network announced Hill's new position will be senior correspondent for the network and senior columnist for the website. The controversial Sage Steele even took a swipe at her colleague Hill. ESPN's Steele faced backlash for criticizing a player who didn't stand for the National Anthem and protestors of the Trump travel ban. Steele reportedly said in an interview that Hill brought the problems on herself. Without mentioning Steele by name Hill responded by twitter. Hill says her policy for saying anything regarding a colleague is she'll say to them in private. She also noted she rarely cares what most of those colleagues' opinions are anyway. Ouch!
Philly's own (and my "nephew" SMILE) Kevin Hart takes his Irresponsible Tour to Atlantic City. The real deal is Hart's wife Eniko. She's marching forward with her own sense of humor by appearing on a hilarious video with her husband declaring she's on tour with him to keep him out of trouble. Kevin stood up to the cheating/blackmail scandal and marched forward himself. Check out the couple in Hart's tour trailer on my Patty TV page on WDASFM.com. Hart's Irresponsible Tour comes to The Boardwalk Hall in Atlantic City Saturday, May 26th.
Essence magazine released the Essence Music Festival schedule and lineup for 2018. The festivities return to New Orleans July 5th through the 8th. Performers include Janet Jackson, Jilly from Philly Jill Scott, Philly's own The Roots, Snoop Dogg, Erykah Badu, Teddy Riley's New Jack Swing Experience (Wreckx-N-Effect, Blackstreet and Guy), Idris Elba, Fantasia, Miguel and Mary J. Blige with more acts to be announced. All the details are on Essence.com
American Idol, with Lionel Richie, Luke Bryan and Katy Perry in the judges seats and Ryan Seacrest as host, returns to TV on its new network ABC, Sunday, March 11th at 8pm.
Lee Daniels' dramas Empire and Star return to FOX Wednesday nights March 28th beginning at 8pm.
The Black-ish spinoff Grown-ish has been renewed. Actress Yara Shahidi's character, "Zoey Johnson," and her adventures in college is a hit on the ABC's cable network Free Form. Grown-ish airs Wednesdays at 8pm on Free Form, and will return for a second season later this year.
Also renewed is Oscar winner Angela Bassett, Peter Krause (Sports Night; Parenthood) and Connie Britton (Spin City; Friday Night Lights) in FOX's drama 911. The show about ambulance driving rescue workers is a hit and airs Wednesday at 9pm on FOX. Meanwhile, look for Bassett in Marvel Comics big screen adaption of Black Panther, with Chadwick Boseman, to be released February 16th. Speaking of Black Panther, Co-star and Oscar Winner Lupita Nyong'o is on the cover of the Hollywood Reporter. Along with Boseman, Bassett and Nyong'o, Black Panther co-stars Michael B. Jordan, Danai Gurira, Daniel Kaluuya, Forest Whitaker, Phylicia Rashad and Sterling K. Brown, and is directed by Ryan Coogler. Marvel Comics' Black Panther hits the big screen Friday, February 16th, and has already set a new record in advanced tickets sales through Fandango. If you can't wait until February, check out the trailer on my Patty TV page on WDASFM.com.
Yours Truly is in the latest edition of Philadelphia Magazine. I'm talking about life after stroke. Much progress has been made in terms of my health, but a few goals to go as well. Philadelphia magazine is out now.
We remember NBA Player and Philly Native Rasual Butler and his wife former American Idol Contestant and Singer Leah LaBelle. Both were tragically killed in a car crash this week. Leah LaBelle appeared on the 2003-2004 season of Idol and made it to the semi-finals. She eventually signed with L.A. Reid and Epic Records and released the R&B song Sexify, though her album was not released. Bulter grew up in the Point Breeze section of South Philly. After graduating from Roman Catholic High, he had a stand out college basketball career at La Salle University. He then moved on to a 13-year NBA career as a small forward and shooting guard with The Miami Heat, Chicago Bulls, Washington Spurs and, finally, with the San Antonio Spurs. Rasual Butler, 38, and wife Leah LaBelle, 31, were together when the crash happened. Police are still investigating.
The Oscar nominations are out and Get Out made a huge showing. The unlikely box office blockbuster, which returns to theaters this weekend, broke all kinds of records when it grossed over $250 million at the box office last year. It took nods for Best Picture (Jordan Peele and his producing partners), Actor (Daniel Kaluuya), Screenplay and Directing (both for Peele, his debut in both positions). Also nominated Denzel Washington (Best Actor) for Roman J. Israel, Esq.; Supporting Actress nods for Mary J. Blige in Mudbound and Octavia Spencer in The Shape Of Water. Dee Rees and Virgil Williams are nominated for Best Adapted Screenplay for Mudbound, and Dee is the first African American woman to ever be nominated in that category. NBA Legend Kobe Bryant has been nominated for Best Animated Short Film (Dear Basketball) as a producer. Mary J. Blige is actually up for two Oscars. She shares her second nomination with Raphael Saadiq and Taura Stinson for Best Original Song (Mighty River from Mudbound). Common is also nominated again this year and is sharing his nomination with Diane Warren for Stand Up For Something taken from the movie Marshall. Jimmy Kimmel hosts the 90th Annual Academy Awards Sunday, March 4th at 8pm on ABC.
UnSung is back. The TV music and Hollywood documentary series will kick off its 12th season after winning the 2018 NAACP Image Award for Outstanding News and Information Program. UnSung returns Sunday, February 19th at 9pm with music group Silk as the season premiere.
Oprah Winfrey is on the cover of March's InStyle magazine, and Winfrey, Writer/Director Ava DuVernay and Actress Storm Reid are all on the cover of the newly black-owned Essence magazine talking about their new movie A Wrinkle In Time. Oprah calls the movie the new Wizard Of Oz. A Wrinkle In Time hits theaters March 9th.
Gabrielle Union in a mother's day thriller from Will Packer. Union reunites with Producer Will Packer for the thriller Breaking In. Union plays a woman whose father passes and when she brings her two children to his highly secured home, the horror begins. When a group of criminals break into the home, hold the kids hostage and try to extort Union's character, her race against time becomes recuing her children. Breaking In opens Mother's Day, May 13th.
Speaking of the big game, Comedienne/Actress Tiffany Haddish will team up with Groupon for a Superbowl ad. Haddish went on Jimmy Kimmel Live and told a hilarious story about using Groupon for a swamp tour with Will and Jada Smith. The story went viral and the team up seemed totally natural. Haddish says she should have already been with Groupon since she's been talking about them publicly for years. Her ad will air in the fourth quarter of the Superbowl. She's also signed a two-year first look deal with HBO. Undoubtly, a comedy special will probably be part of the deal. Haddish also has the TBS show The Last OG with Tracy Morgan coming in April, and she has the movie Night School coming in September with Kevin Hart.
Uncensored is coming to TV One in February. The series explores the intimate lives and past experiences of some of today's hottest talent from various media. Tiffany Haddish, Tamar Braxton, Rick Ross, NeNe Leakes and Nia Long are a few of the featured subjects. The biography series Uncensored debuts Sunday, February 18th at 10pm on TV One.
Taraji P. Henson is currently starring in the action-thriller Proud Mary, which will reunite Color Purple Co-stars Danny Glover and Margaret Avery. Soon will be Henson's Tyler Perry-psychological-thriller Acrimony, co-starring Tika Sumpter (The Haves & Have Nots; Ride Along), opening March 30th.
Singer/Songwriter/Actor John Legend is coming to TV as…..Jesus Christ. The Oscar winner will star in NBC's revival of the legendary Broadway smash Jesus Christ Superstar. Legend is no stranger to Broadway works as he won a Tony for producing the revival of August Wilson's Jitney. This project is the latest in the live revivals of movies and plays by network television. Rock Legend Alice Cooper will also be in the cast. Jesus Christ Superstar will air live on NBC Easter Sunday, April 1st.
Larenz Tate (Love Jones; Dead Presidents) and Erica Ash (The Real Husbands Of Hollywood) will host The 2018 Trumpet Awards, the celebration of African American achievement and history in society, airing Sunday, February 11th at 9pm on Bounce TV.
Check out who's coming to town:
February, Friday, the 16th and Saturday, 17th – Mary J. Blige in concert at the Borgata in Atlantic City. Tickets and more information at WWW.THEBORGATA.COM!
Saturday, February 10th – Chris Tucker comedy show at The Sugar House Casino, 1001 Delaware Avenue. For tickets or for more information log onto sugarhousecasino.com
Sunday, February 11th – The Juice Crew Reunion starring Big Daddy Kane, MC Shan, Marley Marl, Tuff Crew, Kool G. Rap and MC Lyte at The Electric Factory on 7th Street between Willow and Spring Garden. Tickets and more information log onto Ticketmaster.com
Saturday, February 17th – Kem/Fantasia/Avant in concert at The Boardwalk Hall in Atlantic City. For tickets and information log onto ticketmaster.com.
Sunday, February 18th – The Soul Jam starring The Stylistics, The Delfonics, Blue Magic and Bloodstone At The Tropicana in Atlantic City. Tickets and more information log onto Tropicana.net
Saturday, February 24th – Katt Williams in concert at The Boardwalk Hall in Atlantic City. For tickets and information log onto boardwalkhall.com/events.
Friday, March 9th – Keith Sweat in concert at The Liacouras Center, Broad & Montgomery. Tix are on sale now and for more information log onto Liacourascenter.com/events.
Thursday, April 5th – Patti LaBelle in concert at The Academy Of Music, 240 South Broad Street, between Locust and Manning Streets. For tickets and information log onto kimmelcenter.org
Friday, April 13th – The Philly Spring Jam starring Ginuwine, Tank, Brandy, Teddy Riley & Guy and 112 at the Liacouras Center and for more information log onto Liacourascenter.com/events.
Friday, May 11th – The 10th Anniversary Of Platinum Shows' Mother's Day Music Festival begins with the Friday concert starring Singers Joe and Stokley (from Mint Condition) kick things off Friday, May 11th at Caesar's Palace in Atlantic City. For Ticket information log onto ppshows.com.
Saturday, May 12th – The 10th Anniversary Of Platinum Shows' Mother's Day Music Festival with the Saturday concert starring Maxwell, Gladys Knight and Vivian Green at the Boardwalk Hall in Atlantic City. For Ticket information log onto ppshows.com.
Sunday, May 13th – The 10th Anniversary Of Platinum Shows' Mother's Day Music Festival concludes with the Sunday concert starring The Whispers and L.J. Reynolds & The Dramatics at The Tropicana in Atlantic City. For Ticket information log onto ppshows.com.
If you would like more info please email me at PattyJackson@iHeartMedia.com or check me out with my midday reports at 10:30am, 12:30pm and 2:30pm during the day.
Each week Patty Jackson gives us the latest celebrity news and entertainment gossip in The 4-1-1 every Thursday on WDAS Read more


I rolled over in my bed groggily and looked at the alarm clock. The red numbers looked blurry until my eyes cleared and I could see that the screen said 7:00 AM. As I walked clumsily to the bathroom to get dressed for the day I wondered what the me from eleven years ago, or even 3 years ago would think of me waking up this late on a workday, a workday in the summer no less.
I own an air conditioning contracting business and when I started the business eleven years ago it would have never occurred to me that waking up at 7AM in the busiest time of the year would be an option. Waking up at 5AM was normal and for a while I even experimented with waking up as early as 3AM to get things done before the demands of the workday began.
That changed, and at first I wasn't sure why. Things were good at work, great in fact. Profits are up, volume is up and I get to work with a group of nearly 50 fantastic people every day. I found things that used to excite me didn't anymore and things that used to upset me didn't bother me.
When I started in the HVAC trade I enjoyed learning the business, figuring out how things worked, keeping up on the newest technology and going home every day knowing that I had fixed something and learned something new. But time passes, you grow weary, or maybe just forget to "stop and smell the roses". I was sitting in my office and it hit me, I used to really enjoy the WORK ITSELF. It was never just a means to an end, it wasn't that I was just building a career, a family or a business, I was doing something I enjoyed.
Some people can do business for the sake of business, or can go to work everyday simply for the paycheck; successful small business owners are not usually those people.
Forgive me if this is the most obvious statement in the history of business articles, but if you stop doing what you enjoy, you won't be as good at your job.
If you own a large florist shop it may not make sense for you as the owner to do nothing but make arrangements all day, but you can do one every day. Give yourself 30 minutes a day to reconnect with the work that you enjoy and what made you start the business in the first place- and not only will you enjoy the business more, you will find your perspective begins to change.
You may find you aren't as quick as you once were at doing your craft, some of the details may even escape your memory, but the cool part is that if you have been successful you won't need to be as productive/efficient as you once were; the pressure isn't on, so take your time and let your experience fuel your creativity and out of that creativity you will find new avenues that may benefit your entire organization.
Business can feel like a game of "Should"- I should do this, I shouldn't do that, We should be past this by now, etc..
It's your business, it's your career, it's your life. If you stop to smell the roses either literally or figuratively that is your decision, and it's one you are unlikely to regret.
So I am waking up at 5:30 again and giving classes to my staff about the trade, reading articles about what's new and I even started a podcast devoted to air conditioning and technicians to share what I know and what I'm learning.
I feel alive again at work.
Make space to do the things that made you come alive when you first started, don't do it for the affirmation or the ego of it, don't do it to "show you've still got it", do it because you want to. That's reason enough.


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Q: LUIS gives 403 error "Out of call volume quota." when testing in portal with using Authoring Resource F0 As I am trying to test LUIS app, it is throwing 403 error with this message:"Out of call volume quota for LUIS.Authoring F0 pricing tier. Please retry after 9 days. To increase your call volume switch to a paid tier."
I am using the Azure Authoring resource key tier F0 which has a limit of 1 million/month, 5/second. Across all apps, we have made only 2310 API calls this month and we have not made more than 5 per second. I am not using the prediction resource key as it has a monthly limit of 10 thousand.
I got a similar error couple of days ago in which the message was to try again after 11 days but it started working later for a day and then again got the error to try after 9 days. It makes me wonder if there is any other limit on a daily or weekly basis or is this due to some other issue?
I read similar posts here but couldn't figure out a resolution. It would be great if anyone can share any insights on how to resolve this issue.

A: The 1 million/month is for authoring transactions only. These would be the programmatic calls to get intent lists, add applications, train applications, etc. This doesn't apply to actually testing the application through your in-portal testing. The limit for testing predictions with the authoring key is only 1,000/month. You can just create a free tier prediction resource and associate it to your LUIS app which will upgrade you to 10,000/month.
Microsoft has good documentation on LUIS Azure resources if you need additional information.

A: Thanks @billoverton for the response. I was using Power Automate to test and the power automate connector for LUIS was only accepting the authoring key, which is a known issue. So instead, I have directly called the API and used the prediction key and that has fixed the issue.

redpajama_set_name: RedPajamaStackExchange


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Silwan/City of David
Silwan / City of David
The Green Belt
The Cable Car
The old Muslim cemetery in Mamilla
Community Excavations
The Old City - Q&A
Frequently asked questions about the Old City	Read more >>
Emek Shaveh is an Israeli NGO working to prevent the politicization of archaeology in the context of the Israeli-Palestinian conflict.
Press Release: Decision to Approve Elad's "Kedem Compound" Plan in Silwan Awakens Fear of Extreme Politicization of Planning and Building Institutions
The decision of the National Planning and Building Council made public just a few hours ago, to approve construction of the Kedem Compound on an area of some 15,000 sq. m., could have been predicted in advance, from the moment that the Justice Minister forced the National Planning and Building Council to hold a rehearing on the plan. This rehearing took place yesterday (Tuesday, March 22) and the sense throughout was that it was only a formality that would ultimately result in approval of the plan to establish the ideological mall of Elad and the Nature and Parks Authority adjacent to the Old City walls at the entrance to the village of Silwan. This decision is a key step in strengthening the settlement of Silwan and will inflict severe damage on important archeological remains discovered there and deal a harsh blow to the multi-cultural nature of Jerusalem.
Throughout the hearing, blatant indications of political support for the plan were patently evident. Council Chairman Mr. Avigdor Yitzhaki, for example, pointed out the great importance of the building. Mr. Yitzhaki mentioned several times that it was necessary to erect a building "that would protect visitors from the sun," and stated that in his opinion, the Regional Committee knows every meter and street of Jerusalem and therefore, its professional opinion should be trusted. Director General of the Justice Ministry, Ms. Emi Palmor, also stated her positive regard for the program. During the hearing, members of Silwan were removed from the hearing room, and the attorney representing them left in protest against the dismissive attitude towards the needs of the Palestinian residents of the city.
In the previous round, which took place in May 2015, prior to the elections, the appeals committee of the Planning and Building Council decided to drastically reduce the size of the building. After a two-day hearing, members of the committee formulated a 120-page case for their decision. In this round, at the rehearing, members of the National Council devoted only four hours to discussing the grandiose plan, and their decision was substantiated in a one-page document only.
The decision to build a mall-sized building on top of Jerusalem antiquities, for the benefit of a political organization, arouses the fear that he planning institutions have been politicized in the extreme, as well as the fear that it will be destructive to Jerusalem as a World Heritage Site. Instead of leaving antiquities exposed and accessible to all, the planning committees, bowing to political pressure, decided to approve unprecedented damage to the antiquities and to transfer these relics of historical heritage to Elad, an organization with a right wing nationalist ideology. The Antiquities Authority's support of this project testifies to the inability of this governmental organ to protect Jerusalem's past and preserve it for the general public today and for the benefit of future generations.
Elad Givati Parking lot Kedem Compound Silwan Back to top
עמק שווה - Emek Shaveh.


well done!! I'm still waiting for an offer, but starting to feel doubtful..although I have only just sent my written assignment off so not too worried yet.
Received an offer for Law only a few minutes ago!!! Conditional, for 39 points in IB and 19 in my high levels. Only got an email from UCAS, Portico status still says "Currently being processed by Admissions". Wow, I still can't believe it!!!
Congratulation!!! The first rush of reading the offer must have been awesome!
I got an offer for law as well! I am an IB student too. Are you firming it?
I'm still waiting for LSE to reply (hopefully tomorrow or the week after that) and then I'll see. Although I don't know how I'll make up my mind about that decision if it comes to that Where are you from?
I would wait too for LSE!!! I am from Greece, you?
what was your bmat score if you don't mind me asking?
The ones who have an offer for maths: what's with the step support? Is it free? Are there video lessons? Or documents?
However, I did perform very well at both the UCL and Cambridge interviews, which is the most important factor in my opinion - so don't lose hope if your bmat score isn't the best it could've been!
Arts & sciences offer holder?
Maths offer holder as of about 4pm today. Application sent in on December 12th.
History (SSEES) offer holder as of 6 pm, Moscow time, January 23rd. Applied January 15th.
Well done!!! Where do you think you'll firm?
I'm still waiting for LSE, Bath and Cardiff before I decide. Can't really start thinking about firming anything yet until I have all of the offers that I'm going to get.


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You are at:Home»Sections»E-commerce»Clicks to bricks: Yuppiechef to open physical stores in Gauteng
Clicks to bricks: Yuppiechef to open physical stores in Gauteng
By Suren Naidoo	 14 June 2019 No Comments
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Pioneering South African online kitchen and homeware retailer Yuppiechef is opening more physical stores in an omni-channel retail push. The Cape-based group announced on Tuesday that it will be opening its first stores in the Johannesburg market later this month.
Two new stores are set to open — at Mall of Africa and Sandton City — on Friday, 28 June, extending Yuppiechef's physical store foray. The group opened its first bricks-and-mortar store in Cape Town two years ago, after its founders, Andrew Smith and Shane Dryden, realised that physical stores would give the group access to a bigger market beyond just online.
Many digital natives, including Smith, once claimed that e-commerce would take over and spell the death of physical retail and even shopping centres, but have now become omni-channel retail evangelists.
We believe the future of retail is 'omni-channel', which means a combination of physical stores and e-commerce
"We believe the future of retail is 'omni-channel', which means a combination of physical stores and e-commerce," says Smith, who is CEO of Yuppiechef.
The opening of Yuppiechef's first two stores in Gauteng will bring the number of its physical stores to six. Its current stores are all in the Western Cape, with the most recent opening at the V&A Waterfront in October last year.
Smith says there are plans for further expansion through its physical store roll-out. "For now, we're focusing on getting our stores in Johannesburg up and running. We'd love to have Yuppiechef stores all around South Africa in the coming years."
Yuppiechef is a privately run group that was launched by Smith and Dryden as an online retail start-up in 2006 out of Smith's home in Cape Town. Today it is a multimillion-rand enterprise with an employee base that is set to hit 150 this year.
'As easy as possible'
Asked about Yuppiechef's move into bricks-and-mortar stores, Smith remarks: "A large portion of the South African market is not comfortable shopping online, or at least not for everything on every occasion. Our goal is to make the shopping experience as easy as possible for our customers, whether they want to shop online or in-store. They can browse actual products on the shelf, speak to a shop assistant and then choose to have their purchase delivered directly to their home. It's a seamless integration of the physical and online world."
He adds: "We no longer consider Yuppiechef an e-commerce company, but just a retailer, and so our competitors are traditional retail chains, general online stores and specialist kitchen retailers. There is an overlap in some of the brands we sell, but we consider our specialist kitchen and home range the biggest in the country."
According to Smith, Yuppiechef sells more than 10 000 kitchen and home products that have been carefully curated for quality and uniqueness.
"Our top sellers always include the classics like KitchenAid mixers, Wüsthof knives, Sagenwolf cookware and our League of Beers craft beer cases. With a shift toward environmental consciousness, the rising stars are re-usable straws (silicon and stainless options), Zoku water bottles and SodaStream," he says.
On Yuppiechef's target market, Smith says its customer base is diverse in both location and demographic, with deliveries across South Africa as well as a few neighbouring countries. "Our customers are anyone who enjoys making and sharing food at home. We have had strong support from Cape Town and Gauteng, which is why we started our physical store expansion here."
Where can we expect Yuppiechef in the next decade?
The move to omni-channel retail by both traditional retailers like Pick n Pay and online retailers like Yuppiechef has gained momentum globally
Smiths enthuses: "Ten years ago we had only just moved out of my lounge, so 10 years from now feels like a long way away. A successful transition into a national omni-channel retailer is our immediate goal, and we'll see where our journey takes us from there."
Moneyweb reported last month about SA's largest online retailer Takealot's roll-out of its first 30 physical collection points around the country. It remains to be seen whether the Naspers-owned business will eventually start rolling out fully fledged physical stores too, or perhaps turn some of its pick-up points into stores.
The move to omni-channel retail by both traditional retailers like Pick n Pay and online retailers like Yuppiechef has gained momentum globally over the last few years. According to a report published in December by commercial real-estate services group JLL in the US, e-commerce retailers plan to open 850 physical stores in the next five years.
"The clicks-to-bricks retailers' expansion plans demonstrate the value these brands place on having a physical presence with which to engage shoppers," notes JLL.
This article was originally published on Moneyweb and is used here with permission
Andrew Smith Shane Dryden Takealot top Yuppiechef
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Possibly a contentious unbeliever will maintain that we worshiping images in our churches are convicted of praying to lifeless idols. Far be it from us to do this. Faith makes Christians, and God, who cannot deceive, works miracles. We do not rest contented with mere colouring. With the material picture before our eyes we see the invisible God through the visible representation, and glorify Him as if present, not as a God without reality, but as a God who is the essence of being.
In fact, the whole earth is a living icon of the face of God.


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You are here: Home » Profile » Jamaicans World Wide Vote Elaine Thompson Herah 2021 Person of the Year
Elaine Thompson Herah
[SOUTH FLORIDA] – Jamaican athlete Elaine Thompson Herah was voted the Person of the Year for 2021 at the Best of Jamaica Awards. They are sponsored by the world's top website and media platform focused on Jamaican and Caribbean news, travel, food, and culture. The Best of Jamaica is the platform's biggest event of the year. It is all about what its readers and staff vote as the best Jamaica has to offer.
The Best of Jamaica 2021 is based on the results of two surveys: The Best of Jamaica Abroad and The Best of Jamaica in Jamaica. Both ask respondents for what – and who – they think is the best. The surveys seek responses from members of Jamaican Diaspora communities in major cities in the USA, UK and Canada.
The results were announced on December 10, 2021, on the Jamaicans.com YouTube, Instagram, and Facebook pages on a show hosted by Jamaican media personalities Jody Ann Gray and Eddy Edwards. Jody Ann Gray enjoys a large fan base for her radio shows in Jamaica and Florida. Eddy Edwards is the executive producer and co-host of Caribbean Riddims, a popular radio program airing on CANEradio.com.
Jamaicans' choice for 2021 Person of the Year is the sprinter and five-time Olympic champion Elaine Sandra-Lee Thompson Herah. She was born in Banana Ground, Manchester, Jamaica in 1992. Unlike other champion runners, Thompson Herah was not a stand-out athlete in her early years. Over time, with the help of her disciplined and religious family, her own strong work ethic, and some world-class scouting, she overcame her early challenges to become one of the best and most recognized athletes in the world.
Thompson Herah credits her grandmother for her start on the journey to becoming a world-class athlete. One of her chores growing up was to go to the shops and bring back the items her grandmother requested. She would memorize the list of items her grandmother told her to get at the shops. Then she'd run back home as fast as she could to watch her favorite cartoons and movies on television.
She competed for Christiana High School and Manchester High School during her teen years. Taking fourth place in Class Two 100 meters at the Jamaica ISSA Grace Kennedy Boys and Girls Championships in 2009. She clocked 12.01 seconds for the distance.
After high school, Paul Francis, the brother of MVP Track Club head coach Stephen Francis, recruited her to compete for the University of Technology. Thompson Herah said his tough-love approach was instrumental in getting better performances from her. He pointed out to her that she was not attaining the times she should be. Additionally, he counseled her to be less serious, not to be afraid of people, and to "shake it up."
Competitive Results
Her performance markedly improved under his coaching. In 2013, she ran a personal best of 11.41 seconds at the Gibson Replay and took second place at the Jamaica Intercollegiate Championships. She then ran the first leg of the 4×100-meter relay for Jamaica at the Central American and Caribbean Championships. Making a major contribution to Jamaica's win of the gold medal with a time of 43.58 seconds. She won her first intercollegiate title in 2014 with a fifth-place finish at the national championships, clocking 11.26 seconds. Her breakthrough moment came in 2015 at the Utech Classic. She ran the 100 meters in under 11 seconds for the first time, as well as achieving a personal best of 21.66 seconds in the 200-meter event. Thompson Herah ranked 30th in the world with a personal best time in the 100 meters of 10.84 seconds at this event.
Olympic Triumphs
Injuries kept her from repeating her phenomenal performance at the 2016 Rio Olympics where she won gold medals in the 100 meters and the 200 meters. However, she became more dedicated to her goal than ever. She ran seven 100-meter races in 2020 and turned in times of less than 11 seconds in five of them. She won two Diamond League competitions that year and had a season's best time in the 200 meters of 22. 19 seconds.
At the Tokyo Olympics in 2021, Thompson-Herah took gold in the women's 100-meter final, equaling the second-fastest time in history and setting a Jamaican national record and an Olympic record with a time of 10.62 seconds. Running the 200 meters, she equaled her personal best time of 21.66 seconds in the semi-finals and won gold in the final. Achieving a personal best of 21.53 seconds. At the Prefontaine Classic in August 2021, Thompson Herah was the winner in the 100-meter event with a new personal best of 10.54 seconds. This was the second-fastest time in women's sprint history.
Thompson-Herah holds a bachelor's degree in foodservice management and culinary arts. She often prepares dishes that reflect her Caribbean heritage. These include yellow yam, which fellow Olympic champion Usain Bolt has cited as a key reason for the superior sprint performance of Jamaica's athletes. Thompson-Herah is also interested in fashion and favors glamorous, elegant clothes when she is not on the track. Thompson-Herah is married to Derron Herah, a former athlete and coach.


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