This beautiful Gulf view Majestic Sun 701B condo at Seascape Resort is located directly across from the beach on the seventh floor and sleeps a total of ten. This condo offers amazing Gulf views from all common areas including the master bedroom! Wake up and enjoy your morning coffee while taking in the beautiful sights of the sugar white sands and Summers breeze from your private balcony. With this rental, you are provided with two beach chairs and an umbrella in your condo that you can take to the beach, in lieu of spending $40 a day using the local beach service operator. For added convenience, our office is onsite, on the ground floor of the B building in Majestic Sun should you need anything.
Upon entering this three bedroom, three bath condo, you will find a comfortable living area where you can admire the beauty of the Gulf or watch TV from the HD flat-screen. The sleeper sofa in the living area is brand new and can comfortably accommodate up to two guest. The kitchen is fully equipped with brand new stainless steel appliances you would need for a home cooked meal. The dining area and can seat up to six comfortably. Enjoy your meals at the dining table with beautiful views of the Gulf that the Emerald Coast has to offer! The master king suite offers beautiful Gulf views with access to the private balcony, an HD flat-screen TV, private bath and can sleep two. The second bedroom is a queen suite with an HD flat-screen TV, private bath and can sleep two. The third bedroom can sleep up to four guest with two double sized beds and an attached bathroom. This unit conveniently offers a laundry room and sleeps a total of ten.
"We stayed here for a week in May 2019 and loved this property. It is nicely updated. It is beautifully decorated, convenient to the outlet mall and many wonderful restaurants, has a gorgeous ocean and beach view, and was very spacious. Destin Palms was very easy to rent through, and made communication a breeze. This property is a 5 star for sure!"
Thank you so much for your feedback. I am so glad you and your family enjoyed your stay. We look forward to seeing you next year!
"Overall, a great stay at the Majestic Sun in March, 2019. The unit was consistent with the photos and was very clean. The view from the balcony was great. Location is a short walk to the beach - just cross the street. I would definitely recommend this unit for anyone that is bringing their family on vacation to Miramar Beach."
"Michael, thank you so much for feedback from your recent spring stay! I am so glad to hear you found the condo beautiful and clean when you arrived! You were truly a pleasure to host! Here at Destin Palms Vacations we strive to help our guests experience the very best of the Emerald Coast. We look forward to welcoming you back to Majestic Sun again in the near future! See you at the beach! Destin Palms Management"
"Very clean and comfortable. We enjoyed our stay. The only downside was the extreme water pressure in the showers. I suggest variable shower heads to help regulate the pressure. Overall a good experience and we would come back. "
"Terry, you and your family were a pleasure to host. Thank you so much for sharing your experience with us. We look forward to your next stay with us! Best Regards , Destin Palms Management "
"Very happy with the entire visit. Given early checkin and the unit was as nice as it appeared online."
"Thank you for your review Richard! I am so happy to hear that you and your family had a great stay at Majestic Sun. You and your guests were certainly a pleasure to host, and we welcome you all to come back and relax with us here in Destin anytime! Sincerely, Destin Palms Management"
"It was a very nice place to stay. We always stay st the Majestic Sun when we go to Destin. This one had three bedroom and three full bathrooms. We live the balcony . "
"Thank you for your wonderful review Stacy! It is excellent to hear that you and your family made some great memories here at the beach and enjoyed this spectacular rental property at Majestic Sun. Once again, we thank you for choosing to stay with us here at Destin Palms and we look forward to having the pleasure of your company again in the near future! Best Regards, Destin Palms Management"
"We came to Miramar Beach in July 2018 and had a wonderful time. The property was in fantastic shape- clean, comfortable and we especially enjoyed the view from the balcony and the kids loved the huge TV with Netflix. There was plenty of space for our group of seven. We really appreciated having three bathrooms. We spent much of our time on the beach but also adventured to Crab Island for the first time and rented a pontoon. The Whale's Tale is within walking distance and had great food with an awesome view. We also loved Louis Louis- best food we've had in Destin. However, we can't come to Destin without visiting Landsharks (pizza and wings). We caught a movie at Destin Commons and enjoyed shopping at Silver Sands Outlet (which is so close). "
"Thank you Bethany! We sincerely appreciate you taking the time to share your review and are delighted that you had a memorable stay at our condo in Majestic Sun. Louis Louis is one of my personal favorites as well, especially the crab cakes-YUM. Next time you need some relaxation here on the gorgeous Gulf Coast we hope you consider Destin Palms Vacations again to take care of you and your family! Sincerely, Destin Palms Management"
"Clearly, the owners of this condo wanted to go above and beyond for their patrons. All the furniture is high quality and added to the comfort of our stay. Seriously, they spared no expense and every area was accounted for.
Another aspect we enjoyed about the condo we're the colorful walls that matched the blue skies we enjoyed all week long!
Next, having 3 beds and 3 baths took care of us. We never felt crowded and 3 baths made getting ready for our day or evening super easy. Also, if you stay in for eating, all the dishes, cutlery, and cooking ware needed are included which made life easy.
Last but not least, (most important actually) the viewwww!! The view was worth the price of admission. I'd stay for the view alone but everything else mentioned went a long way in making our stay that much more enjoyable!
We were very fortunate to find this place!
"Thank you for taking the time to share your experience with us Daniel! It is so wonderful to hear that you and your family found Majestic Sun 701B had plenty of space and enjoyed the furnishings and amazing view of this spectacular rental property. You and your guests were certainly a pleasure to host, and we welcome you all to come back and relax with us here in Destin anytime! Sincerely, Destin Palms Management"
"It was a great week. The condo was perfect for my family. It was very clean and a great location. The rental agency was very helpful. They take exceptional care of their properties. I would recommend this rental agency others."
"Thank you for your review Flossie! I am delighted to read that you and your family had a wonderful stay at Majestic Sun. We hope to welcome you and your family to stay with us here in paradise again soon! Sincerely, Destin Palms Management"
"We enjoyed our stay at Majestic Sun 701B. The unit is spacious and has an excellent view. The location is close to everything and the walk to the beach was very short. We definitely will stay here again. "
"Great ocean views! Condo directly across the street from the beach. Initially, I was a little worried we might be bothered by the street noise but it wasn't a problem. The roar of the ocean muffled out the noise. At night the street was very quiet and we could sleep with the sliding glass door open to hear the ocean. The condo was very comfy and clean. Only suggestion would be a bistro table for balcony. It would of been nice to of been able to eat a meal on the balcony. Only plastic chairs with small side tables to sit our drinks on were provided. We had a great time at Miramar Beach. Beaches were beautiful! We would recommend the SeaBlaster boat ride. It was one of the highlights of our trip. "
"Thank you for your positive review! Thanks for that great idea, I will send it over to my property manager. We hope to see you again soon. :) Best regards, Destin Palms Mangement"
"The condo was very spacious, clean, and updated. The decor was bright and very comfortable. The furniture was all nice and new. The TV in the living area is huge and perfect for a family movie. The kitchen was well stocked with pots, pans, cookie sheets, etc. The unit includes nice beach chairs that are a pleasure to relax in on the beach. The view is gorgeous and Majestic Sun is in a great location, with beautiful pool amenities. Super close to the outlet mall and tons of great restaurants. Key-less entry made coming and going from the unit a breeze. We had a wonderful week in 701B and highly recommend."


The Prime Minister, Mr Narendra Modi today lauded the efforts of Haryana Chief Minister, Mr Manohar Lal and Uttar Pradesh Chief Minister, Mr Yogi Adityanath in strengthening law and order in both the States.
The Prime Minister was addressing a large gathering after the inauguration of Eastern Peripheral Expressway (Kundli-Ghaziabad-Palwal) Expressway at Baghpat in Uttar Pradesh today.
He said that earlier the criminal after committing crime in one state used to run to other state. However, with the coordinated efforts of the Chief Ministers of both the states, the criminals have now been surrendering, which is resulting in reduction in crime and providing better law and order situation in both the states, he added.


This afternoon, 35 Emerging Leaders across all cohorts from 2014 to 2018 united to discuss the research and implementation of key projects around the topical theme of Access to Essential Medicines, featuring at events around this year's World Congress of Cardiology.
Three different research teams focused on health systems, health policy and implementation presented their findings and discussed and debated the challenges and opportunities to apply their learning across other further initiatives.
The GLOPRO team focus their research on Essential Medicines Lists (EML's) and specifically to petition new medicines on national EMLs specifically in India and Zimbabwe. As a second phase this team want to better understand and map LICs, LMICs and UMICs with no priority CVD essential medicine in EMLs. This research offers so much potential to further expand and consider possible recommendations on how to strengthen processes for improved access to essential medicines.
The HAVE Meds team investigate the affordability and accessibility to essential medicines in Mozambique and using an affordabiltiy scale of less than one day's wage to purchase 1 month's supply of medicines, almost all CVD medications are either unaffordable or unaffordable and unavailable in local context.
The CLUB Meds team described their adherence to medication research in Nigeria. Using a qualitative study of focus groups and interviews, the team described their collection of research across nine different adherence 'clubs', and the data that can assist patients to adhere to hypertension medication within limited resource settings.


text:
\section{Representations and Complexity}
\label{sec:complexity}

\subsection{Sparse Pseudo-Boolean Functions}

For any $n \in \mathbb{N}$, any $f \colon \{0,1\}^n \to \R$ is called an $n$-variate \emph{pseudo-Boolean} function (PBF). Any $n$-variate PBF $f$ has a unique multi-linear polynomial form\footnote{In line with the literature on pseudo-Boolean optimization, we call the form \eqref{eq:multi-linear-poly} \emph{multi-linear} despite its constant term $c_\emptyset$.}:
\begin{align}
f(y) = \sum_{I \subseteq [n]} c_I \prod_{i \in I} y_i,
\qquad \textnormal{with} \qquad
c \colon 2^{[n]} \to \R.
\label{eq:multi-linear-poly}
\end{align}
We observe that the objective functions of the tree partition problem{} form a class of sparse PBFs whose set $\{I \subseteq [n] \colon c_I \neq 0\}$ of non-zero coefficients is constrained by a tree. This class of functions is formalized in the following definition.

\begin{defn}
An $n$-variate PBF is called \emph{tree-sparse} w.r.t.~a tree $T = (V, [n])$ iff its multi-linear polynomial form \eqref{eq:multi-linear-poly} is such that, for every $I \subseteq [n]$ with $c_I \neq 0$, the set $I$ induces a path in $T$.
The function is called \emph{path-sparse} w.r.t.~a path $P = (V, [n])$ iff it is tree-sparse w.r.t.~$P$.
\end{defn}

\subsection{Set Partitioning}

For any connected subgraph (subtree) $S \subseteq T$, introduce a binary variable $\lambda_S \in \{0,1\}$. By defining the cost $d_S = \sum_{u,v \in V_S} c_{uv}$ for each component $S$, problem \eqref{eq:problem-pbo} can be reformulated as the set partitioning problem
\begin{align}
\min \quad & d^\top \lambda \label{eq:problem-set-partition} \\
\text{s.t.} \quad & \sum_{S : u \in V_S} \lambda_S = 1, \quad \forall u \in V \nonumber \\
& \lambda_S \geq 0, \; \lambda_S \in \Z. \nonumber
\end{align}
Here, the costs $d_S$ account for all pairs of nodes within component $S$ and the constraints ensure that every node is contained in exactly one component of the partition. Note that the number of variables $\lambda_S$ is exponential in the number of leaves of $T$.

\begin{lem}
The vector $y \in \{0,1\}^E$ is a solution of problem \eqref{eq:problem-pbo} w.r.t.\ the tree $T=(V,E)$ and costs $c \colon \tbinom{V}{2} \to \R$ iff the vector $\lambda$ defined by
\begin{align*}
\lambda_S = 1 \iff \forall e \in S : y_e = 1
\end{align*}
for every subtree $S \subseteq T$, is a solution of problem \eqref{eq:problem-set-partition} w.r.t.\ the costs $d_S = \sum_{u,v \in V_S} c_{uv}$.
\end{lem}

\begin{proof}
We set $c_{uu} = 0$ for all $u \in V$. Let $S_u$ denote the component of the partition that contains $u \in V$. The claim follows from
\begin{align*}
\sum_{\{u,v\} \in \mybinom{V}{2}} c_{uv} \prod_{e \in P_{uv}} y_e = & \sum_{u \in V} \sum_{v \in V_{S_u}} c_{uv} \prod_{e \in P_{uv}} y_e \\
= & \sum_{u \in V} \sum_{v \in V_{S_u}} c_{uv} \\
= & \sum_{u \in V} \sum_{\substack{\text{subtree} \\ S \subseteq T : u \in V_S}} \lambda_S \sum_{v \in V_S} c_{uv} \\
= & \sum_{\substack{\text{subtree} \\ S \subseteq T}} \lambda_S \sum_{u,v \in V_S} c_{uv}. \qedhere
\end{align*}
\end{proof}

\subsection{Lifted Multicuts}

In this section, we identify the tree partition problem{} (Def.~\ref{def:problem-main}) as a special case of the minimum cost lifted multicut problem \citep{hornakova-2017} where the underlying graph is a tree. Essentially, this representation is a linearization of the tree partition problem{} with reversed binary encoding.

The \emph{minimum cost lifted multicut problem} w.r.t.\ the tree $T=(V,E)$ and costs $c \colon \tbinom{V}{2} \to \R$ is the combinatorial optimization problem
\begin{align}
\min_{x \in X_T} \sum_{\{u,v\} \in \mybinom{V}{2}} c_{uv} \, x_{uv}, \label{eq:problem-lmp}
\end{align}
where $X_T$ is defined as
\begin{align}
X_{T} & = \Big \{ x \in \{0,1\}^m \; \big| \nonumber \\
& x_{uv} \leq \sum_{e \in P_{uv}} x_e && \hspace{-1em} \forall u,v \in V, \dist(u,v) \geq 2, \label{eq:tree-lmp-path} \\
& x_e \leq x_{uv} && \hspace{-1em} \forall u,v \in V, \dist(u,v) \geq 2, \; \forall e \in P_{uv} \label{eq:tree-lmp-cut} \Big \}.
\end{align}
Problem \eqref{eq:problem-lmp} is the special case of the minimum cost lifted multicut problem as presented by \citet{hornakova-2017} for the specific choice $G = T$ and $G' = (V, \mybinom{V}{2})$, i.e.\ the tree $T$ lifted to the complete graph on $V$ (cf.\ appendix \ref{sec:lifted-multicuts}). Lemma \ref{lem:tree-lmp} states that the tree partition problem{} can be reformulated as a minimum cost lifted multicut problem.
\begin{lem} \label{lem:tree-lmp}
The vector $y \in \{0,1\}^n$ is a solution of problem \eqref{eq:problem-pbo} w.r.t.\ the tree $T=(V,E)$ and costs $c \colon \mybinom{V}{2} \to \R$
iff
the unique $x \in X_T$ such that $x_e = 1 - y_e$ for all $e \in E$ is a solution of problem \eqref{eq:problem-lmp} w.r.t.\ $T$ and the cost function $-c$.
\end{lem}

\begin{proof}
For any distinct pair of nodes $u,v \in V$, we introduce a binary variable $x_{uv} \in \{0,1\}$ via
\begin{align}
x_{uv} = 1 - \prod_{e \in P_{uv}} y_e
\end{align}
which implies
\begin{align}
x_{uv} = 0
& \quad \iff \quad
\forall e \in P_{uv} \colon \ y_e = 1 \nonumber \\
& \quad \iff \quad
\forall e \in P_{uv} \colon \ x_e = 0. \label{eq:vars-transform}
\end{align}
Therefore, we can reformulate problem \eqref{eq:problem-pbo} in terms of the variables $x_{uv}$ by transforming the objective function according to
\begin{align}
c_{uv} \prod_{e \in P_{uv}} y_e & = -c_{uv} \big(1 - \prod_{e \in P_{uv}} y_e \big) + c_{uv} \nonumber \\
& = -c_{uv} \, x_{uv} + c_{uv}.
\end{align}
This leads to the combinatorial optimization problem
\begin{align}
\min_{x \in X_T} \sum_{\{u,v\} \in \mybinom{V}{2}} -c_{uv} \, x_{uv} + c_{uv}, \label{eq:tree-lmp-problem}
\end{align}
where $X_T \subseteq \{0,1\}^m$ captures the relationship \eqref{eq:vars-transform}.
\end{proof}

Note that, since the objective function of \eqref{eq:problem-lmp} is linear, we can replace $X_T$ by its convex hull
\begin{align}
\Xi_T = \conv X_T,
\end{align}
which is called the \emph{lifted multicut polytope} w.r.t.\ $T$. This motivates the study of the structure of $\Xi_T$ in Section \ref{sec:polyhedral}.
We refer to \eqref{eq:tree-lmp-path} and \eqref{eq:tree-lmp-cut} as \emph{path} and \emph{cut} inequalities, respectively, and write $\Theta^0_T$ for the naive linear relaxation of $\Xi_T$, i.e.\ the set of vectors $x \in [0,1]^m$ that satisfy \eqref{eq:tree-lmp-path} and \eqref{eq:tree-lmp-cut}.

\subsection{Complexity}

We now discuss the computational complexity of the tree partition problem{} and the path partition problem{}.
In Lemma~\ref{lem:np-hardness}, we show that the tree partition problem{} is \textsc{np}-hard, even if the tree is a star.
On the contrary, the path partition problem{} is polynomial time solvable \citep{Kernighan1971}.

\begin{lem}
\label{lem:np-hardness}
The tree partition problem{} is \textsc{np}-hard.
It remains \textsc{np}-hard if $T$ is a star.
\end{lem}

\begin{proof}
For a tree-sparse pseudo-Boolean function defined on a star with $n$ leaves, problem \eqref{eq:problem-pbo} is equivalent to the unconstrained binary quadratic program with $n$ variables, which is well-known to be \textsc{np}-hard.
\end{proof}

\section{Conclusion}

We have characterized all trivial facets of the lifted multicut polytope for trees. Additionally, we provide a tighter relaxation compared to the standard linear relaxation by including additional classes of facets. The described facets provide a complete totally dual integral (TDI) description of the associated lifted multicut polytope for paths. This result relates the geometry of this problem to the combinatorial properties of alternative formulations such as the sequential set partitioning problem.

\section{Introduction}

We study the problem of decomposing (clustering) a tree with respect to costs attributed to pairs of nodes, so as to minimize the sum of costs for those pairs of nodes that are in the same component (cluster).
This \emph{tree partition problem{}} is stated rigorously in Def.~\ref{def:problem-main}.
One instance and its solution are depicted in Fig.~\ref{fig:tree-decomposition-costs}.

On the one hand, the tree partition problem{} lacks a complexity of clustering problems for general graphs:
Its feasible set, the set of all decompositions of a tree, is trivial to characterize.
For example, the decompositions of any tree $T = (V, E)$ relate one-to-one to the binary edge labelings $y \in \{0,1\}^E$ that indicate whether neighboring nodes $\{u, v\} \in E$ are in the same component ($y_{uv} = 1$) or distinct components ($y_{uv} = 0$).

On the other hand, the tree partition problem{} exhibits a complexity not commonly found in clustering problems for general graphs:
Its objective function has a multilinear form in the coordinates of $y$ (in the standard unit basis) whose polynomial degree is the length of the longest path in $T$.
This form arises from the fact that any two distinct nodes $\{u, v\} \in \tbinom{V}{2}$ are in the same component iff $y_e = 1$ for all edges $e$ on the unique path $P_{uv}$ in $T$ from $u$ to $v$, that is, iff $\prod_{e \in P_{uv}} y_e = 1$.
Hence, the tree partition problem{} can be stated rigorously as follows:

\begin{defn}
\label{def:problem-main}
For any (finite, simple, undirected) tree $T = (V, E)$ and
any $c \colon \tbinom{V}{2} \to \mathbb{R}$,
the optimization problem \eqref{eq:problem-pbo} is called the instance of the \emph{tree partition problem{}} w.r.t.~$T$ and $c$.
If $T$ is a path, it is also called the instance of the \emph{path partition problem{}} w.r.t.~$T$ and $c$.
\begin{align}
\min_{y \in \{0,1\}^E}
\sum_{\{u,v\} \in \mybinom{V}{2}}
    c_{uv}
\prod_{e \in P_{uv}}
    y_e
\label{eq:problem-pbo}
\end{align}
\end{defn}

\begin{figure}
\centering
\begin{tikzpicture}[shift={(0,-0.0)},xscale=2.5,yscale=2.0]
\draw[draw=mylightblue!65, fill=mylightblue!65] plot[smooth cycle, tension=0.5] coordinates
    {(-0.1, 0) (0.15, 0.45) (0, 1) (0.2, 1.1) (0.55, 0.5) (0.1, -0.1)};
\draw[draw=mylightblue!65, fill=mylightblue!65] plot[smooth cycle, tension=0.5] coordinates
    {(0.9, 0.6) (0.95, 0.85) (1.7, 0.6) (1.6, 0.35)};

\node (0) [style=vertex,fill=white,label=below:{\scriptsize}] at (0, 0) {};
\node (1) [style=vertex,fill=white,label=below:{\scriptsize}] at (0.1, 1.0) {};
\node (2) [style=vertex,fill=white,label=below:{\scriptsize}] at (0.4, 0.5) {};
\node (3) [style=vertex,fill=white,label=above:{\scriptsize}] at (1.0, 0.7) {};
\node (4) [style=vertex,fill=white,label=below:{\scriptsize}] at (1.6, 0.5) {};

\draw (0) edge[base] node[left=4,above=-2] {\footnotesize $2$} (2);
\draw (1) edge[base] node[left] {\footnotesize $-1$} (2);
\draw (2) edge[cut-edge,base] node[above] {\footnotesize $-2$} (3);
\draw (3) edge[base] node[above] {\footnotesize $-1$} (4);

\draw[lifted] (0) edge[bend left=40] node[left,black] {\footnotesize $-3$} (1);
\draw[cut-edge,lifted] (1) edge[bend left=30] node[left=5, below=-1,black] {\footnotesize $1$} (3);
\draw[cut-edge,lifted] (0) edge[bend right=35] node[left=6,above=-4,black] {\footnotesize $1$} (3);
\draw[cut-edge,lifted] (0) edge[bend right=50] node[above,black] {\footnotesize $2$} (4);
\draw[cut-edge,lifted] (1) edge[bend left=50] node[above,black] {\footnotesize $-2$} (4);
\draw[cut-edge,lifted] (2) edge[bend right=25] node[above=6,right=2,black] {\footnotesize $3$} (4);
\end{tikzpicture}
\caption{Depicted above is an instance of the tree partition problem{} with costs associated to pairs of nodes.
Edges of the tree are depicted as straight bold lines (dotted or solid).
Additional pairs of nodes are marked by thin blue curves (dotted or solid).
The solution decomposes the tree into two components, depicted with a shaded background, leaving some pairs of nodes in the same component (solid lines and curves) and some pairs of lines in distinct components (dotted lines and curves).}
\label{fig:tree-decomposition-costs}
\end{figure}

\vspace{-3ex}

\subsection{Contribution}

In Section~\ref{sec:complexity}, we survey several representations of the tree partition problem{} and discuss its computational complexity.
For the general case and for the special case of the tree being a star, we show that the problem is \textsc{np}-hard. The path partition problem{}, on the other hand, is known to be polynomial time solvable \citep{Kernighan1971}.

In Section~\ref{sec:polyhedral}, we study the lifted multicut polytope \citep{hornakova-2017} associated with the lifted multicut representation of the tree partition problem{}. We characterize several classes of facets for the general case, which in particular yield a complete totally dual integral description (TDI) of the lifted multicut polytope for paths. Our results relate the geometry of lifted multicuts to the combinatorial properties of the tree and path partition problem{}.

\subsection{Notation}

Throughout the paper, we consider $T = (V,E)$ to be a (finite, simple, undirected) tree with $n = \lvert E \rvert$ edges. Let $m = \lvert \mybinom{V}{2} \rvert$ be the number of distinct pairs of nodes in $V$.
For any $\{u,v\} \in \tbinom{V}{2}$, we denote by $P_{uv}$ the unique path in $T$ connecting $u$ and $v$.
Moreover, we denote by $\dist(u,v)$ the distance of $u$ and $v$ in $T$, i.e.\ the length of $P_{uv}$.
To ease notation, we take the liberty of identifying any graph $G = (V_G, E_G)$ with its edge set $E_G$.
Hence, we may write $e \in G$ instead of $e \in E_G$.
For any $k \in \mathbb{N}$, we denote by $[k] := \{1, \dotsc, k\}$ the set of integers from $1$ through $k$.

\section{Polyhedral Geometry}
\label{sec:polyhedral}

\begin{figure*}
\subfloat[]{%
\begin{tikzpicture}[shift={(0,-0.35)},scale=1.3]
\node (0) [style=vertex,fill=white,label=below:{\scriptsize $0$}] at (0, 0.2) {};
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\node (6) [style=vertex,fill=white,label=below:{\scriptsize $6$}] at (1.7, 1.5) {};

\draw (0) edge[base] node[right] {} (1);
\draw (1) edge[base] node[right] {} (2);
\draw (1) edge[base] node[right] {} (3);
\draw (3) edge[base] node[right] {} (4);
\draw (2) edge[base] node[right] {} (5);
\draw (2) edge[base] node[right] {} (6);

\draw[-] (4) edge[bend left=50,lifted] node[below,black] {$x_{46}$} (6);
\draw[-] (0) edge[bend left=30,lifted] node[above] {} (3);
\draw[-] (0) edge[bend right=40,lifted] node[above,black] {$x_{05}$} (5);
\end{tikzpicture}
\label{fig:graphs-tree}
}
\hfill
\subfloat[]{%
\begin{tikzpicture}[scale=1]
\node (1) [style=vertex,fill=white,label=left:{\scriptsize $1$}] at (0, 0) {};
\node (2) [style=vertex,fill=white,label=right:{\scriptsize $3$}] at (2, 0) {};
\node (3) [style=vertex,fill=white,label=above:{\scriptsize $2$}] at (1, 1.73) {};
\node (4) [style=vertex,fill=white,label=below:{\scriptsize $0$}] at (1, 0.575) {};

\draw[-] (1) edge[bend right=50,lifted] node[above,black] {$x_{13}$} (2);
\draw[-] (2) edge[bend right=50,lifted] node[right,black] {$x_{23}$} (3);
\draw[-] (1) edge[bend left=50,lifted] node[left,black] {$x_{12}$} (3);
\draw (3) edge[base] (4);
\draw (1) edge[base] node[above=4] {$x_{01}$} (4);
\draw (2) edge[base] (4);
\end{tikzpicture}
\label{fig:graphs-star}
}
\hfill
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\begin{tikzpicture}[shift={(0,0.25)},scale=1]
\node (0) [style=vertex,fill=white,label=below:{\scriptsize $0$}] at (0, 0) {};
\node (1) [style=vertex,fill=white,label=below:{\scriptsize $1$}] at (1, 0) {};
\node (2) [style=vertex,fill=white,label=below:{\scriptsize $2$}] at (2, 0) {};
\node (3) [style=vertex,fill=white,label=below:{\scriptsize $3$}] at (3, 0) {};

\draw (0) edge[base] node[right] {} (1);
\draw (1) edge[base] node[above] {$x_{12}$} (2);
\draw (2) edge[base] node[above] {} (3);
\draw[-] (0) edge[bend left=60,lifted] node[above=3,right=10,black] {$x_{02}$} (2);
\draw[-] (1) edge[bend right=55,lifted] node[below,black] {$x_{13}$} (3);
\draw[-] (0) edge[bend left=80,lifted] node[above,black] {$x_{03}$} (3);
\end{tikzpicture}
\label{fig:graphs-path}
}
\hfill
\subfloat[]{%
\begin{tikzpicture}[scale=1.3]
\draw[base] plot [smooth] coordinates {(1,0) (1.2,0.1) (1.45,0.05)};
\draw[base] plot [smooth] coordinates {(2.0,-0.05) (2.3,-0.1) (2.5,0)};
\node (dots) at (1.75,0) {$\dotso$};
\node (0) [style=vertex,fill=white,label=left:{\scriptsize $u$}] at (0.5, 0.5) {};
\node (1) [style=vertex,fill=white,label=below:{\scriptsize $\vec{u}(v)$}] at (1, 0) {};
\node (2) [style=vertex,fill=white,label=below:{\scriptsize $v$}] at (2.5, 0) {};

\draw (0) edge[base] node[right] {} (1);
\draw[-] (0) edge[bend left=70,lifted] node[above=3,right=10,black] {} (2);
\draw[-] (1) edge[bend left=60,lifted] node[below,black] {} (2);
\end{tikzpicture}
\label{fig:graphs-triangle}
}
\caption{\ref{fig:graphs-tree}: A tree with additional lifted edges (thin blue curves) between non-neighboring nodes corresponding to long range variables. \ref{fig:graphs-star}: A star with three leaves lifted to the complete graph. \ref{fig:graphs-path}: A path of length $3$ lifted to the complete graph. \ref{fig:graphs-triangle}: The node $\vec{u}(v)$ is the first node on the path $P_{uv}$.}
\label{fig:graphs}
\end{figure*}

In this section, we establish polyhedral results for the lifted multicut polytope $\Xi_T$ for trees. We characterize all trivial facets and offer a tighter outer relaxation of $\Xi_T$. In Section~\ref{sec:paths}, we show that our results yield a complete totally dual integral (TDI) description of the lifted multicut polytope for paths. This result relates the combinatorial properties of the sequential set partitioning problem to the geometry of the minimum cost lifted multicut problem for paths. The rather technical proofs for our claims are deferred to appendix \ref{sec:proofs}.

\subsection{Canonical Outer Relaxation} \label{sec:tree-relax}
We give another simple relaxation of $\Xi_T$ that is at least as tight as the naive linear relaxation $\Theta^0_T$. To this end, define $\vec{u}(v)$ to be the first node on the path $P_{uv}$ that is different from both $u$ and $v$ (cf.\ Fig.\ \ref{fig:graphs-triangle}) and consider the polytope
\begin{align}
\Theta^1_T & = \Big \{ x \in [0,1]^m \; \big | \nonumber \\
& x_{uv} \leq x_{u,\vec{u}(v)} + x_{\vec{u}(v),v} && \forall u,v \in V, \; \dist(u,v) \geq 2, \label{eq:tree-path}  \\
& x_{\vec{u}(v),v} \leq x_{uv} && \forall u,v \in V, \; \dist(u,v) \geq 2 \label{eq:tree-cut} \Big \}.
\end{align}

This description is canonical in the sense that it only considers a quadratic number of node triplets, namely those which feature two neighboring nodes and an arbitrary third node. The following lemma states that $\Theta^1_T$ is indeed an outer relaxation of $\Xi_T$ which is at least as tight as $\Theta^0_T$.

\begin{lem} \label{lem:tree-polytope}
It holds that $\Xi_T \subseteq \Theta^1_T \subseteq \Theta^0_T$.
\end{lem}

\subsection{Trivial Facets}
\label{sec:tree-facets}

It can be characterized exactly which inequalities of \eqref{eq:tree-path}, \eqref{eq:tree-cut} and $0 \leq x_{uv} \leq 1$ define facets of $\Xi_T$. These elementary results are closely related to the more general study in \citep{hornakova-2017}.

\begin{lem} \label{lem:tree-path-facets}
The inequality \eqref{eq:tree-path} defines a facet of $\Xi_T$ if and only if $\dist(u,v) = 2$.
\end{lem}

\begin{lem} \label{lem:tree-cut-facets}
The inequality \eqref{eq:tree-cut} defines a facet of $\Xi_T$ if and only if $v$ is a leaf of $T$.
\end{lem}

\begin{lem} \label{lem:tree-box-facets}
For any $u,v \in V,\; u \neq v$, the inequality $x_{uv} \leq 1$ defines a facet of $\Xi_T$ if and only if both $u$ and $v$ are leaves of $T$. Moreover, none of the inequalities $0 \leq x_{uv}$ define facets of $\Xi_T$.
\end{lem}

\subsection{Non-Trivial Facets}

We further present a large class of non-trivial facets of $\Xi_T$. Consider the set of inequalities given by
\begin{align}
x_{uv} + x_{\vec{u}(v),\vec{v}(u)} & \leq x_{u,\vec{v}(u)} + x_{\vec{u}(v),v} \label{eq:tree-square} \\
& \forall u,v \in V,\; \dist(u,v) \geq 3. \nonumber
\end{align}
We shall refer to \eqref{eq:tree-square} as \emph{square} inequalities. As an example consider the graph depicted in Fig.\ \ref{fig:graphs-path} and $u=0$, $v=3$. Any square inequality is valid and facet-defining for $\Xi_T$.

\begin{lem} \label{lem:tree-square-valid}
The inequalities \eqref{eq:tree-square} are valid for $\Xi_T$.
\end{lem}

\begin{lem} \label{lem:tree-square-facets}
The inequalities \eqref{eq:tree-square} define facets of $\Xi_T$.
\end{lem}

\subsection{Facets Related to the Boolean Quadric Polytope}

The argument to show Lemma \ref{lem:np-hardness} also implies that the lifted multicut polytope for a star with $n$ leaves is isomorphic to the \emph{Boolean Quadric Polytope} \citep{Padberg1989} of order $n$. Therefore, insights about the facial structure of the Boolean quadric polytope can be used to derive valid inequalities and facets for the more general polytope $\Xi_T$. In fact, any tree-sparse pseudo-Boolean function repeatedly admits substructures similar to the quadratic case. As this connection is beyond the scope of our paper, we only illustrate it with the following example. Consider the star graph depicted in Fig.\ \ref{fig:graphs-star}. It is known from \citep{Padberg1989} that the ``triangle'' inequality
\begin{align}
x_{01} + x_{23} \leq x_{12} + x_{13} \label{eq:triangle}
\end{align}
is valid and facet-defining for $\Xi_T$. Observe that, similarly, for the tree in Fig.\ \ref{fig:graphs-tree}
\begin{align}
x_{02} + x_{56} \leq x_{05} + x_{06}
\end{align}
holds true, but in this case $x_{02}, x_{05}$ and $x_{06}$ correspond to cubic instead of quadratic terms of the associated pseudo-Boolean function.

\subsection{Complete TDI Description for Paths}
\label{sec:paths}

Suppose the node set $V = \{0, \dotsc, n\}$ is linearly ordered and consider the path $P = (V,E)$, where the edge set is given by $E = \big\{ \{i,i+1\} \mid i \in \{0, \dotsc, n-1\} \big \}$. We show that the facets arising from \eqref{eq:tree-path} - \eqref{eq:tree-square} yield a complete description of $\Xi_P$.
For this section, we consider only paths of length $n \geq 2$, since for $n=1$ the polytope $\Xi_P = [0,1]$ is trivial. Let $\Theta^{\mathrm{Path}}_P$ be the polytope defined by
\begin{align}
\Theta^{\mathrm{Path}}_P = \Big \{ & x \in \R^m \; \big | \nonumber \\
 & x_{0n} \leq 1, \label{eq:path-box} \\
& x_{in} \leq x_{i-1,n} & \hspace{-7em} \forall i \in \{1, \dotsc, n-1\}, \label{eq:path-cut-right} \\
& x_{0i} \leq x_{0,i+1} & \hspace{-7em} \forall i \in \{1, \dotsc, n-1\}, \label{eq:path-cut-left} \\
& x_{i-1,i+1} \leq x_{i-1,i} + x_{i,i+1} & \hspace{-7em} \forall i \in \{1, \dotsc, n-1\}, \label{eq:path-triangle} \\
& x_{j,k} + x_{j+1,k-1} \leq x_{j+1,k} + x_{j,k-1} \nonumber \\
& \qquad \qquad \forall j,k \in \{0, \dotsc, n\}, \: j < k-2 \label{eq:path-square} \Big \}.
\end{align}

Note that the system \eqref{eq:path-box} - \eqref{eq:path-square} consists precisely of those trivial and square inequalities which we have shown to define facets of $\Xi_P$. We first prove that $\Theta^{\mathrm{Path}}_P$ indeed yields an outer relaxation of $\Xi_P$.

\begin{lem} \label{lem:path-polytope}
It holds that $\Xi_P \subseteq \Theta^{\mathrm{Path}}_P \subseteq \Theta^1_P$.
\end{lem}

As our main result, we prove that $\Theta^{\mathrm{Path}}_P$ is in fact a complete description of $\Xi_P$.
To derive this, we utilize the notion of total dual integrality. For an extensive reference on this subject we refer the reader to \citep{Schrijver1986}.

\begin{defn}
A system of linear inequalities $Ax \leq b$ with $A \in \Q^{k \times m}, \; b \in \Q^k$ is \emph{totally dual integral} (TDI) if for any $c \in \Z^m$ such that the linear program $\max \{ c^\top x \mid Ax \leq b \}$ is feasible and bounded, there exists an integral optimal dual solution.
\end{defn}

Total dual integrality is an important concept in polyhedral geometry as it serves as a sufficient condition on the integrality of polyhedra according to the following fact.

\begin{fact}[\citep{Edmonds1977}]
\label{fact:tdi-integral}
If $Ax \leq b$ is totally dual integral and $b$ is integral, then the polytope defined by $Ax \leq b$ is integral.
\end{fact}

\begin{thm}
\label{thm:path-tdi}
The system \eqref{eq:path-box} - \eqref{eq:path-square} is totally dual integral.
\end{thm}

\begin{cor}
\label{cor:complete-description}
It holds that $\Xi_P = \Theta^{\mathrm{Path}}_P$.
\end{cor}

\begin{proof}
This is immediate from Lemma \ref{lem:path-polytope}, Fact  \ref{fact:tdi-integral} and Theorem \ref{thm:path-tdi}.
\end{proof}

\begin{rem}
The constraint matrix corresponding to the system \eqref{eq:path-box} - \eqref{eq:path-square} is in general \emph{not} totally unimodular. A minimal example is the path of length $4$.
\end{rem}

The set partitioning representation of the path partition problem w.r.t.\ $P$ admits a quadratic number of variables and a linear number of constraints (opposed to a quadratic number of constraints in the description of $\Xi_P$). This representation corresponds to the dual program of the last problem in the proof of Theorem \ref{thm:path-tdi}, cf.\ appendix \ref{sec:proofs}. Here, the integrality constraint need not be enforced, since the constraint matrix is totally unimodular.

\section{Related Work}

\subsection{Pseudo-Boolean Optimization}
The optimization of pseudo-Boolean functions plays an important role in machine learning, for instance, in MAP-inference for computer vision models. The general problem can be reduced to the quadratic case \citep{Boros2002, Boros2012}, which is responsible for the \textsc{np}-hardness of the problem. The combinatorial polytope associated with the linearization of quadratic pseudo-Boolean functions was extensively studied by \citet{Padberg1989}. Computational approaches to pseudo-Boolean optimization based on the roof duality bound \citep{Hammer1984,Kahl2012} have been quite successful in practice \citep{Rother2007}.

\subsection{Correlation Clustering and Multicut Polytopes}
Decompositions of graphs into an unknown number of components based on (dis-)similarities between \emph{neighboring} nodes is referred to as weighted correlation clustering. It has been studied for the complete graph \citep{Bansal2004} as well as general graphs \citep{Demaine2006}. As any decomposition of a graph is characterized by the mathematical notion of a multicut, the study of multicuts is closely related to correlation clustering. The combinatorial polytopes associated with the multicuts of a graph have been studied, among others, most notably by \citet{Groetschel1989, Chopra1993, Deza1997}. The more general case where correlations between \emph{non-neighboring} nodes are taken into account as well has been introduced by \citet{hornakova-2017}.

\subsection{Sequential Set Partitioning}
Set partitioning problems where the elements are assumed to adhere to a linear order have been studied by \citet{Kernighan1971}, who devises a dynamic programming algorithm that essentially solves a shortest path problem in a directed acyclic graph. The corresponding integer linear programming formulation admits a totally unimodular constraint matrix \citep{Joseph1997}.

\section{Proofs for Section \ref{sec:polyhedral}}
\label{sec:proofs}

\paragraph{Proof of Lemma \ref{lem:tree-polytope}.}

We show first that $\Xi_T \subseteq \Theta^1_T$. For this purpose, let $x \in \Xi_T$. If $x$ violates \eqref{eq:tree-path}, then $x_{uv} = 1$ and $x_{u,\vec{u}(v)} = x_{\vec{u}(v),v} = 0$. This contradicts the fact that $x$ satisfies all cut inequalities w.r.t.\ $\vec{u}(v),v$ and the path inequality w.r.t.\ $u,v$. If $x$ violates \eqref{eq:tree-cut}, then $x_{\vec{u}(v),v} = 1$ and $x_{uv} = 0$. This contradicts the fact that $x$ satisfies all cut inequalities w.r.t.\ $u,v$ and the path inequality w.r.t.\ $\vec{u}(v),v$. Hence, $x$ satisfies \eqref{eq:tree-path}-\eqref{eq:tree-cut} and therefore $x \in \Theta^1_T$.

Now, we show that $\Theta^1_T \subseteq \Theta^0_T$. Let $x \in \Theta^1_T$. We need to show that $x$ satisfies all path and cut inequalities. Let $u,v \in V$ with $\dist(u,v) \geq 2$. We proceed by induction on $\dist(u,v)$. If $\dist(u,v) = 2$, then the path inequality is directly given by \eqref{eq:tree-path}, while the two cut inequalities are given by \eqref{eq:tree-cut} for the two possible orderings of $u$ and $v$. If $\dist(u,v) > 2$, then the path inequality is obtained from \eqref{eq:tree-path} and the induction hypothesis for the pair $\vec{u}(v),v$, since $\dist(\vec{u}(v),v) = \dist(u,v) - 1$. Similarly, for any edge $e$ on the path from $u$ to $v$, we obtain the cut inequality w.r.t.\ $e$ by using the induction hypothesis and \eqref{eq:tree-cut} such that (w.l.o.g.) $e$ is on the path from $\vec{u}(v)$ to $v$. Hence, $x \in \Theta^0_T$.

\paragraph{Proof of Lemma \ref{lem:tree-path-facets}.}

First, suppose $\dist(u,v) = 2$. Then $P_{uv}$ is a path of length $2$ and thus chordless in the complete graph on $V$. Hence, facet-definingness follows directly from \citep[Thm 4]{hornakova-2017}. Now, suppose $\dist(u,v) > 2$ and let $x \in \Xi_T$ be such that \eqref{eq:tree-path} is satisfied with equality. We show that this implies
\begin{align}
x_{uv} + x_{\vec{u}(v),\vec{v}(u)} = x_{u,\vec{v}(u)} + x_{\vec{u}(v),v}. \label{eq:square-equality}
\end{align}
Then the face of $\Xi_T$ induced by \eqref{eq:tree-path} has dimension at most $m - 2$ and hence cannot be a facet. In order to check that \eqref{eq:square-equality} holds, we distinguish the following three cases. If $x_{uv} = x_{u,\vec{u}(v)} = x_{\vec{u}(v),v}$, then all terms in \eqref{eq:square-equality} vanish. If $x_{uv} = x_{u,\vec{u}(v)} = 1$ and $x_{\vec{u}(v),v} = 0$, then $x_{\vec{u}(v),\vec{v}(u)} = 0$ and $x_{u,\vec{v}(u)} = 1$, so \eqref{eq:square-equality} holds. Finally, if $x_{uv} = x_{\vec{u}(v),v} = 1$ and $x_{u,\vec{u}(v)} = 0$, then \eqref{eq:square-equality} holds because $x_{\vec{u}(v),\vec{v}(u)} = x_{u,\vec{v}(u)}$ by contraction of the edge $u,\vec{u}(v)$.

\paragraph{Proof of Lemma \ref{lem:tree-cut-facets}.}

First, suppose $v$ is not a leaf of $T$ and let $x \in \Xi_T$ be such that \eqref{eq:tree-cut} is satisfied with equality. Since $v$ is not a leaf, there exists a neighbor $w \in V$ of $v$ such that $P_{\vec{u}(v),v}$ is a subpath of $P_{\vec{u}(v),w}$. We show that $x$ additionally satisfies the equality
\begin{align}
x_{uw} = x_{\vec{u}(v),w} \label{eq:tree-cut-equality}
\end{align}
and thus the face of $\Xi_T$ induced by \eqref{eq:tree-cut} cannot be a facet. There are two possible cases: Either $x_{uv} = x_{\vec{u}(v),v} = 1$, then $x_{uw} = x_{\vec{u}(v),w} = 1$ as well, or $x_{uv} = x_{\vec{u}(v),v} = 0$, then $x_{uw} = x_{vw} = x_{\vec{u}(v),w}$ by contraction of the path $P_{uv}$, so \eqref{eq:tree-cut-equality} holds.

Now, suppose $v$ is a leaf of $T$ and let $\Sigma$ be the face of $\Xi_T$ induced by \eqref{eq:tree-cut}. We need to prove that $\Sigma$ has dimension $m-1$. This can be done explicitly by showing that we can construct $m-1$ distinct indicator vectors $\chi_{w,w'}$ for $w, w' \in V$ as linear combinations of elements from the set $S = \{x \in \Xi_T \mid x_{uv} = x_{\vec{u}(v),v}\}$.
In fact this construction is analogous to the one used in the proof of Theorem 7 in \citep{hornakova-2017} where the authors derive the dimension of the lifted multicut polytope $\dim(\Xi_{GG'}) = \lvert E' \rvert$. The difference here is that the vector $\mathbbm{1} - \chi_{uv} \notin S$, so we have to distinguish between $x \in S$ with $x_{uv} = x_{\vec{u}(v),v} = 0$ and $x \in S$ with $x_{uv} = x_{\vec{u}(v),v} = 1$ in order to show $\chi_{u,\vec{u}(v)} \in \lin(S)$ and then $\chi_e \in \lin(S)$ for all other $e \neq  \{u,v\}, \{\vec{u}(v),v\}$.

\paragraph{Proof of Lemma \ref{lem:tree-box-facets}.}

We apply the more general characterization given in \citep{hornakova-2017} to our special case. The nodes $u,v \in V$ are a pair of $w$-$w'$-cut-vertices for some vertices $w,w' \in V$ (with at least one being different from $u$ and $v$) if and only if $u$ or $v$ is not a leaf of $V$. Thus, the claim follows from \citep[Thm 8]{hornakova-2017}. The second assertion follows from \citep[Thm 9]{hornakova-2017} and the fact that we lift to the complete graph on $V$.

\paragraph{Proof of Lemma \ref{lem:tree-square-valid}.}

Let $x \in \Xi_T$ and suppose that either $x_{u,\vec{v}(u)} = 0$ or $x_{\vec{u}(v),v} = 0$ for some $u,v \in V$ with $\dist(u,v) \geq 3$. Then, since $x$ satisfies all cut inequalities w.r.t.\ $u,\vec{v}(u)$, respectively $\vec{u}(v),v$, and the path inequality w.r.t.\ $\vec{u}(v), \vec{v}(u)$, it must hold that $x_{\vec{u}(v),\vec{v}(u)} = 0$. Moreover, if even $x_{u,\vec{v}(u)} = 0 = x_{\vec{u}(v),v}$, then, by the same reasoning, we have $x_{uv} = 0$ as well. Hence, $x$ satisfies \eqref{eq:tree-square}.

\paragraph{Proof of Lemma \ref{lem:tree-square-facets}.}

Let $\Sigma$ be the face of $\Xi_T$ induced by \eqref{eq:tree-square}. We need to prove that $\Sigma$ has dimension $m-1$, which can be done explicitly by showing that we can construct $m-1$ distinct indicator vectors $\chi_{w,w'}$ for $w, w' \in V$ as linear combinations of elements from the set $S = \{x \in \Xi_T \mid x_{uv} + x_{\vec{u}(v),\vec{v}(u)} = x_{u,\vec{v}(u)} + x_{\vec{u}(v),v}\}$.
Again, this construction is very technical and analogous to the proof of Theorem 7 in \citep{hornakova-2017} about the dimension of the lifted multicut polytope.

\paragraph{Proof of Lemma \ref{lem:path-polytope}.}

First, we show that $\Xi_P \subseteq \Theta^{\mathrm{Path}}_P$. Let $x \in \Xi_P$, then $x$ satisfies \eqref{eq:path-box} and \eqref{eq:path-triangle} by definition. Suppose $x$ violates \eqref{eq:path-cut-right}, then $x_{in} = 1$ and $x_{i-1,n} = 0$. This contradicts the fact that $x$ satisfies all cut inequalities w.r.t.\ $i-1,n$ and the path inequality w.r.t.\ $i,n$. So, $x$ must satisfy \eqref{eq:path-cut-right} and, by symmetry, also \eqref{eq:path-cut-left}. It follows from Lemma \ref{lem:tree-square-valid} that $x$ satisfies \eqref{eq:path-square} as well and thus $x \in \Theta^{\mathrm{Path}}_P$.

Next, we prove that $\Theta^{\mathrm{Path}}_P \subseteq \Theta^1_P$. To this end, let $x \in \Theta^{\mathrm{Path}}_P$.
We show that $x$ satisfies all inequalities \eqref{eq:tree-cut}. Let $u,v \in V$ with $u < v - 1$. We need to prove that both $x_{u+1,v} \leq x_{uv}$ and $x_{u,v-1} \leq x_{uv}$ hold. For reasons of symmetry, it suffices to show only $x_{u+1,v} \leq x_{uv}$. We proceed by induction on the distance of $u$ from $n$. If $v = n$, then $x_{u+1,n} \leq x_{un}$ is given by \eqref{eq:path-cut-right}. Otherwise, we use \eqref{eq:path-square} for $j=u$ and $k=v+1$ and the induction hypothesis on $v+1$:
\begin{align*}
x_{uv} + x_{u+v,v+1} & \geq x_{u+1,v} + x_{u,v+1} \\
& \geq x_{u+1,v} + x_{u+1,v+1} \\
\implies x_{uv} & \geq x_{u+1,v}.
\end{align*}
It remains to show that $x$ satisfies all inequalities \eqref{eq:tree-path}. Let $u,v \in V$ with $u < v - 1$. We proceed by induction on $\dist(u,v) = u-v$. If $\dist(u,v) = 2$, then \eqref{eq:tree-path} is given by \eqref{eq:path-triangle}. If $\dist(u,v) > 2$, then we use \eqref{eq:path-square} for $j=u$ and $k=v$ as well as the induction hypothesis on $u,v-1$, which have distance $\dist(u,v) - 1$:
\begin{align*}
x_{uv} + x_{u+1,v-1} & \leq x_{u+1,v} + x_{u,v-1} \\
&  \leq x_{u+1,v} + x_{u,u+1} + x_{u+1,v-1} \\
\implies  x_{uv} & \leq x_{u,u+1} + x_{u+1,v}.
\end{align*}
Hence, $x \in \Theta^1_P$, which concludes the proof.

\paragraph{Proof of Theorem \ref{thm:path-tdi}}

We rewrite system \eqref{eq:path-box} - \eqref{eq:path-square} more compactly in the following way. Introduce two artificial nodes $-\infty$ and $\infty$ where we associate $-\infty$ to any index less than $0$ and $\infty$ to any index greater than $n$. Moreover, we introduce variables $x_{ii}$ for all $0 \leq i \leq n$ as well as $x_{-\infty,i}$ and $x_{i, \infty}$ for all $1 \leq i \leq n-1$ and finally $x_{-\infty, \infty}$. Now, the system \eqref{eq:path-box} - \eqref{eq:path-square} is equivalent to the system
\begin{align}
& x_{j,k} + x_{j+1,k-1} \leq x_{j+1,k} + x_{j,k-1} \nonumber \\
& \qquad \qquad \forall j,k \in \{-\infty, 0, \dotsc, n, \infty \}, \: j \leq k-2 \label{eq:path-4cycle}
\end{align}
given the additional equality constraints
\begin{align}
x_{ii} & = 0 \quad \forall 0 \leq i \leq n, \label{eq:path-vertices} \\
x_{-\infty, i} & = 1 \quad \forall 1 \leq i \leq n-1, \\
x_{i, \infty} & = 1 \quad \forall 1 \leq i \leq n-1, \\
x_{-\infty, \infty} & = 1. \label{eq:path-inf-cut}
\end{align}
Let the system defined by \eqref{eq:path-4cycle} - \eqref{eq:path-inf-cut} be represented in matrix form as $Ax \leq a, \; Bx = b$. Note that $\Theta^{\mathrm{Path}}_P$ is non-empty and bounded. Thus, to establish TDI, we need to show that for any $c \in \Z^{m + 3n}$ the dual program
\begin{align}
\min \{ a^\top y + b^\top z \mid A^\top y + B^\top z = c, y \geq 0\} \label{eq:dual-prog}
\end{align}
has an integral optimal solution. Here, the $y$ variables, indexed by $j,k$, correspond to the inequalities \eqref{eq:path-4cycle} and the $z$ variables, indexed by pairs of $i, -\infty$ and $\infty$, correspond to the equations \eqref{eq:path-vertices}  - \eqref{eq:path-inf-cut}. Then, the equation system $A^\top y + B^\top z = c$ translates to
\begin{align}
y_{i-1,i+1} + z_{i,i} & = c_{i,i} \label{eq:dual-d1} \\
y_{i-1,i+2} - y_{i-1,i+1} - y_{i,i+2} & = c_{i,i+1} \label{eq:dual-d2} \\
y_{i-1,\ell+1} - y_{i-1,\ell} - y_{i,\ell+1} + y_{i,\ell} & = c_{i,\ell} \label{eq:dual-d3} \\
- y_{-\infty,i+1} + y_{-\infty,i} + z_{-\infty,i} & = c_{-\infty,i} \label{eq:dual-left} \\
- y_{i-1,\infty} + y_{i,\infty} + z_{i,\infty} & = c_{i,\infty} \label{eq:dual-right} \\
\phantom{\ +} y_{-\infty,\infty} + z_{-\infty,\infty} & = c_{-\infty,\infty}, \label{eq:dual-inf}
\end{align}
where \eqref{eq:dual-d1} - \eqref{eq:dual-d3} hold for all $0 \leq i < i+1 < \ell \leq n$ and \eqref{eq:dual-left}, \eqref{eq:dual-right} hold for all $1 \leq i \leq n-1$. Observe that \eqref{eq:dual-d1} includes only $y$ variables with indices of distance $2$, \eqref{eq:dual-d2} couples $y$ variables of distance $3$ with those of distance $2$ and \eqref{eq:dual-d3} couples the remaining $y$ variables of any distance $d > 3$ with those of distance $d-1$ and $d-2$. Hence, any choice of values for the free variables $z_{ii}$ completely determines all $y$ variables. This means we can eliminate $y$ and reformulate the dual program entirely in terms of the $z$ variables, as follows. It holds that
\begin{align*}
0 \leq y_{i-1,i+1} & = c_{i,i} - z_{i,i} \qquad \forall 0 \leq i \leq n, \\
0 \leq y_{i-1,i+2} & = c_{i,i+1} + c_{i,i} + c_{i+1,i+1} - z_{i,i} - z_{i+1,i+1} \\
& \qquad \qquad \forall 0 \leq i < i+1 \leq n
\end{align*}
and thus, by \eqref{eq:dual-d3},
\begin{align*}
0 \leq y_{i-1,\ell+1} = \sum_{i \leq j \leq k \leq \ell} c_{j,k} - \sum_{k=i}^{\ell} z_{k,k}
\end{align*}
for all $0 \leq i \leq \ell \leq n$. Substituting the $y$ variables in \eqref{eq:dual-left} - \eqref{eq:dual-inf} yields the following equivalent formulation of \eqref{eq:dual-prog}:
\begin{align}
\min \quad & z_{-\infty,\infty} + \sum_{i=0}^n z_{-\infty,i} + z_{i,\infty} \label{eq:dual-red} \\
\text{s.t.} \quad & \sum_{k=i}^\ell z_{k,k} \leq \sum_{i \leq j \leq k \leq \ell} c_{j,k} \qquad \forall 0 \leq i \leq \ell \leq n \nonumber \\
& z_{-\infty,i} + z_{i,i} = c_{-\infty,i} + \sum_{0 \leq j \leq i} c_{j,i} \quad \forall 1 \leq i \leq n-1 \nonumber \\
& z_{i,\infty} + z_{i,i} = c_{i,\infty} + \sum_{i \leq k \leq n} c_{i,k} \qquad \forall 1 \leq i \leq n-1 \nonumber \\
& z_{-\infty,\infty} - \sum_{k=1}^n z_{k,k} = c_{-\infty,\infty} - \sum_{0 \leq j \leq k \leq n} c_{j,k}. \nonumber
\end{align}
The variables $z_{-\infty,i}, \; z_{i,\infty}$ and $z_{-\infty,\infty}$ occur only in a single equation each. Furthermore, the matrix corresponding to the inequality constraints satisfies the ``consecutive-ones'' property. Therefore, the constraint matrix of the whole system is totally unimodular, which concludes the proof.

\paragraph{Path Partition as Sequential Set Partitioning}

For each $0 \leq i \leq \ell \leq n$, let
\begin{align*}
d_{i,\ell} = \sum_{0 \leq i \leq j \leq k \leq n} c_{j,k},
\end{align*}
then taking the dual of problem \eqref{eq:dual-red} and simplifying yields the problem
\begin{align}
\min \quad & d^\top \lambda \label{eq:ssp} \\
\text{s.t.} \quad & \sum_{0 \leq i \leq k \leq \ell \leq n} \lambda_{i,\ell} = 1, \quad \forall 0 \leq k \leq n \nonumber \\
& \lambda \geq 0. \nonumber
\end{align}
Each variable $\lambda_{i,\ell}$ corresponds to the component containing nodes $i$ to $\ell$. Problem \eqref{eq:ssp} is precisely the sequential set partitioning formulation of the path partition problem{} as used by \citet{Joseph1997}.

\section{Lifted Multicuts of General Graphs}
\label{sec:lifted-multicuts}

For any connected graph $G = (V,E)$,
any supergraph $G' = (V, E')$ with $E' = E \charfusion[\mathbin]{\cup}{\cdot} F$
and any $c: E' \to \mathbb{R}$
the instance of the \emph{minimum cost lifted multicut problem} w.r.t.~$G$, $G'$ and $c$ is the binary linear program
\begin{align}
\min_{x \in X_{GG'}} \sum_{e \in E'} c_e \, x_e \label{eq:lmp-problem}
\end{align}
with
\begin{align}
X_{GG'} & = \Big \{ x \in \{0,1\}^{E'} \; \big | \nonumber \\
& \forall C \in \text{cycles}(G) \, \forall \hat{e} \in C : x_{\hat{e}} \leq \sum_{e \in C \setminus \{\hat{e}\}} x_e, \label{eq:lmp-cycle} \\
& \forall uv \in F \, \forall P \in uv\text{-paths}(G) : x_{uv} \leq \sum_{e \in P} x_e, \label{eq:lmp-path} \\
& \forall uv \in F \, \forall C \in uv\text{-cuts}(G) : 1 - x_{uv} \leq \sum_{e \in C} 1 - x_e \label{eq:lmp-cut} \Big \}.
\end{align}
The inequalities \eqref{eq:lmp-cycle}, \eqref{eq:lmp-path} and \eqref{eq:lmp-cut} are referred to as \emph{cycle}, \emph{path} and \emph{cut} inequalities, respectively.
The convex hull
\begin{align}
\Xi_{GG'} = \conv X_{GG'} \label{eq:lifted-multicut-polytope}
\end{align}
of $X_{GG'}$ in $\mathbb{R}^{E'}$ is called the \emph{lifted multicut polytope} w.r.t.~$G$ and $G'$.
It was shown by \citet{hornakova-2017} that $\Xi_{GG'}$ has dimension $\lvert E' \rvert$.

For any $x \in X_{GG'}$, the set $M := \{e \in E \,|\, x_e = 1\}$ of those edges of the graph $G$ that are labeled 1 is a so-called \emph{multicut} of $G$.
Hence, there exists a decomposition of $G$ such that $M$ is precisely the set of those edges that span across distinct components.
In addition, the set $M' := \{e \in E' \,|\, x_e = 1\}$ of those edges $\{u,v\} = e \in E'$ of the graph $G'$ that are labeled 1 is a multicut of $G'$ lifted from the multicut $M$ of $G$.
A multicut $M'$ of $G'$ lifted from $G$ makes explicit for all pairs $\{u,v\} \in E'$ of nodes (not only those neighboring in $G$) whether $u$ and $v$ are in the same component in the decomposition of $G$.

redpajama_set_name: RedPajamaArXiv


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A POLICE officer has collapsed and died on duty after taking unwell while attending an emergency call at a hotel.
The officer has been asked to respond to a 999 call with a colleague on Monday night when he became ill.
They had driven to the Newhouse Premier Travel Inn in Lanarkshire to deal with the incident before an ambulance had to be called for the 55-year-old constable.
He was rushed to nearby Wishaw General Hospital but died a short time later.
His death is not being treated as suspicious and is not related to the incident he was attending. It is thought to have been caused by a medical condition.
The constable worked for Police Scotland's Q Division which covers the Lanarkshire area.
A source said: "The officers had been on patrol in the area and got a call to attend at the hotel.
"They went in and started speaking to the people involved but the officer then complained to his partner that he felt unwell and went outside.
A Police Scotland spokeswoman said: "Around 11.40pm on Monday, May 3, whilst attending an emergency call at the Newhouse Travel Inn, Edinburgh Road, Motherwell, a 55-year-old serving police officer took unwell and collapsed.
"He was taken by ambulance to Wishaw General Hospital where he sadly died a short time later.


The Millerrite movement suffered what was dubbed the Great Disappointment when both dates in 1845 did not materialize with the return of Jesus. Many had sold all their belongings to wait for Jesus. Even William Miller had to admit he was wrong in his prediction and repented misleadings.(b) However a small group of the Millerrites, which included Ellen G White, refused to admit that they had been led astray. Instead they gravitated around Ellen G White, who had declared herself a prophetess, due to her visions, which conveniently excused the 1844 date by saying rather than returning, Jesus simply had entered the Heavenly Holy of Holies and had begun an "Investigative Judgment".
Today there are the Seventh Day Adventist(SDA) based from California but several offshoots have separated from them claiming to be different or that the SDA church is apostasy or some even claiming Ellen G White to be false. However they all still cling onto Ellen G White's teachings concerning keeping the Sabbath for salvation, Investigative Judgment, the confusing dietary restrictions which is a strict vegan diet (no milks, eggs, or cheese allowed) but allows for pork, and others. These are the Seventh Day Remnant(SDR)(most vocal these days), World's Last Chance(WLC), Berean Church of Free SDA, and the list grows by the day. A simple way to recognize any of them is if they demand Sabbath observance as the Mark of God, otherwise you aren't saved.
c) The Great Controversy:White copied both words and pictures, Proof that Whites "I was shown" visions were even copied!, Proof White copied the pillars of SDA theology She didn't get it From God!
2) Do they teach that Jesus is God and that he died and rose again?
While the Seventh Day Adventist(SDA) and the Seventh Day Remnant(SDR) do teach Jesus died and rose again, a dirty little secret they have is that the Jesus they believe in is really Michael the Archangel. While this might appear to be a minor issue, it actually is a big issue because that makes Jesus not divine and therefore his death is not sufficient to fully save anyone. Jesus always claimed to be God in all the gospels yet they based this one doctrine solely on one verse Jude 1:9 and Ellen G White's opinion that no one but Jesus could face off against the devil.
Yet Michael the archangel, when contending with the devil he disputed about the body of Moses, durst not bring against him a railing accusation, but said, TheLord rebuke thee.
Another secret of the White's estate in California, is that while Ellen G White was telling everyone to not eat meat, she was having chickens prepared for her meals. Also demanding 10% tithing and buying her many books at high prices for the time period made her extraordinarily rich. Only false prophets ever sell their prophecies for money.
For the love of money is the root of all evil: which while some coveted after,they have erred from the faith, and pierced themselves through with many sorrows.
5) Are they speaking of things God has proclaimed not to be known by anyone?
"Soon we heard the voice of God like many waters which gave us the day and hour of Jesus coming...When God spoke the time, He poured upon us the Holy Spirit" (Vol. 1 p. 59).
Here Mrs.White like hundreds date setters to claim to know the day and hour of Jesus return. This flies in direct violation to the Bible proclaiming that it is not to be known by neither man nor angels. No true prophet would be so blatant to go against God's own words.
The SDA have gone as far to quote Ellen G Whites writings over the Bible and have now come out with a new Bible(Clear Word Version) which rewrites myriads of verses that did not back up their Sabbath claims. By far the biggest defining point of SDA and SDR is their observance of the Jewish seventh day Sabbath(Shabat), which is still insisted upon by practicing Jews in Israel today. Ellen G White insisted that Sabbath observance was necessary to stay saved.
EGW: In these things I saw great danger; for if the mind is filled with other things, present truth is shut out, and there is no place in our foreheads for the seal of the living God. This seal is the Sabbath. To Those who are receiving the seal of the living God. —January 31, 1849, paragraph 11.
EGW: "Those who would have the seal of God in their foreheads must keep the Sabbath of the fourth commandment." —7BC 970 (1899).
Unfortunately she missed these following verses that state plainly who the Seal of God really is. No true prophet would have made this mistake.
To understand the how Sabbath was a shadow of Christian's rest from striving in the completed work of Christ, one must understand that belief in Christ atoning death on the cross is completely sufficient to cover ALL our sins(past, present, and future). That is our TRUE sabbath rest.
Here too Ellen G.White falls badly. For a complete list of her failed prophecies refer to Prophecy blunders of Ellen G. White, the SDA prophet! Of course even one according to the Bible is enough to warrant labeling her a false prophet and being stoned to death back in the days of Moses, but they have listed 40 of them. I only pulled out some of the most famous false prophecies she's had.
This statement was made at a 1856 meeting and every one present are now long dead, including Ellen herself in 1915. Obviously, this was a FALSE prophecy. And if this is prophecy she got from the angel is false just what kind of angel was she talking to?
"Thousands have been induced to enlist with the understanding that this war is to exterminate slavery, but now that they are fixed, they find that they have been deceived, that the object of this war is not to abolish slavery, but to preserve it" Vol. 1, pp. 254, 258).
History tells us the US Civil War did indeed abolish slavery. Ellen G White was wrong.
"This nation will yet be humbled into the dust. England is studying whether it is best to take advantage of the present weak condition of our nation and venture to make war on her...When England does declare war, all nations will have one interest of their own to serve, and there will be general war, general confusion"(Vol.1 p.259).
History showed that England not only did not enter the US Civil War but that it had no desire to get involved in any way.
In the year 1850 Bible scholars were teaching of the return of the Jews to Palestine and the restoration of Jerusalem. Here is what Mrs. White said about this: "I also saw that old Jerusalem would never be built up, and that Satan was doing the utmost to lead the minds of the children of the Lord into these things now in the gathering time, to keep them from throwing their interests into the work of the Lord and to cause them to neglect the necessary preparation"(E.W.,p.75).
This is obviously a false prediction, as today on Google earth we can see Old Jerusalem completely rebuilt.
Adventist boast 18 million members worldwide in 2013. It is certainly a sad thing when one person can lead so many astray, by claiming to have visions from God. Ellen G White is definitely a false prophet for she meets 5 out of the 7 indicators. However people are leaving as they start seeing the truth. Here is a List of 100 SDA Pastors fired for rejecting White as prophet!
If you are one of them I want you to learn, as I have, that belief in Jesus is all you need. No amount of good works is good enough to keep you saved, despite what Ellen G White and the churches claim. The Bible is very clear about that. Jesus is about ceasing to our own striving to get to Heaven and getting to Heaven based solely on what Jesus has done alone. That is the good news, no more endless working to get to Heaven. Simply rest in Jesus and He will get you there. You can start by believing in not an angel, but the divine Jesus of the Bible.


Apollo Bay man Christian Feetham has been granted a VISA to stay in Australia, after being held in Maribyrnong Detention Centre since April.
His hometown of Apollo Bay have been rallying since he was at risk of deportation, as President of the Apollo Bay Chamber of Commerce John Gorman reveals Christian is now able to reside in Australia.
Mr Gorman said that Christian has been granted the VISA for 5 years, and he gives credit to Member for Corangamite Sarah Henderson for campaigning on behalf of the people of Apollo Bay, who have been fighting to keep Christian in the country since he was placed in the detention centre earlier this year.
Mr Feetham is a fifth-generation 'Australian' on his mother's side, was born in New Zealand and moved to Australia when he was 3 months old, but several convictions formed the basis of the Federal Government's "bad character" decision, which included the cancellation of his permanent residential visa, and the threat of deportation to New Zealand.


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Lapointe, Pierre Louis. "Bois-des-Filion". The Canadian Encyclopedia, 06 June 2017, Historica Canada. https://www.thecanadianencyclopedia.ca/en/article/bois-des-filion. Accessed 24 April 2019.
Bois-des-Filion owes its name to Antoine Feuillon dit Filion, the French pioneer who settled there in 1684, and to the large maple grove located on the Filion family property.
Bois-des-Filion, Qué, Town, pop 8383 (2006c), 7712 (2001c), inc 1980. Bois-des-Filion is located halfway between Sainte-Thérèse and Terrebonne on the north bank of the Rivière des Mille-Îles, about 20 km from downtown Montréal. First set up as a village under the name of Saint-Maurice de Bois-des-Filion in 1949, the abridged version of the name has been used since its incorporation as a town.
Bois-des-Filion owes its name to Antoine Feuillon dit Filion, the French pioneer who settled there in 1684, and to the large maple grove located on the Filion family property. This residential locality boasts the second-highest illuminated cross in the province of Québec, dedicated to Notre-Dame de Bois-des-Filion. Its winter carnival, L'Hiver en folie (Winter Madness), is celebrated each February.


text:
Skydiving landing area located near the Arbutus Meadows Events Centre in Nanoose Bay. (Google maps)
Victoria man dies after skydiving incident on Vancouver Island
34-year-old had made more than 1,000 jumps
Jul. 6, 2020 12:30 p.m.
A tragic skydiving incident Sunday in Nanoose Bay has claimed the life of a 34-year-old man.
The Victoria man, identified by friends as James Smith, was an experienced parachutist, with more than 1,000 jumps, according to Gord Gauvin, owner of Skydive Vancouver Island.
The incident took placed at approximately 6:30 p.m. on Sunday. Gauvin said the man was attempting an advanced high-speed maneuvre called swooping.
His impact with the ground was "catastrophic", said Gauvin, adding the man succumbed to his injuries immediately.
"The parachutist had a high amount of experience with this maneuvre and was a coach and instructor within the sport of skydiving," Gauvin said in a release. "The skydiving community is a small one on Vancouver Island and together we grieve the loss of our friend and brother."
READ MORE: 98-year-old jumps out of an airplane over Parksville Qualicum Beach
Andy Watson of the B.C. Coroners Service said: "I can confirm we are investigating a death from yesterday – it was related to a skydiving incident. We are investigating to determine how, where, when and by what means a male came to his unexpected death."
Global LNG terminal survey casts doubt on industry as 'safe bet'
Horrifying video shows near head-on collision on Trans Canada

redpajama_set_name: RedPajamaCommonCrawl


text:
THURSDAY, JAN 16 • 7:00
SCIENCE ON SCREEN
Fusing the best of science-themed cinema with engaging post-film conversations, Science on Screen is hosted by North Bay writer David Templeton. Known for his long-running film column Talking Pictures, Templeton is the author of the short novel "Mary Shelley's Body," adapted into a play in 2017, and the upcoming robot-themed science fiction drama "Galatea."
Science on Screen kicks off with Robert Zemeckis' 1997 science-meets-faith drama, the Jodie Foster adventure CONTACT, adapted from the novel by Carl Sagan. The film will be followed by a conversation with Dr. Eugenie Scott, former executive director of the National Center for Science Education, a nonprofit organization supporting teaching of evolutionary science.
This presentation is part of SCIENCE ON SCREEN®, a series pairing film screenings with lively scientific discussions – an initiative of the Coolidge Corner Theatre with major support from the Alfred P. Sloan Foundation.


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text:
const Redis = require('ioredis');
const Promise = require('bluebird');

module.exports = () => new Promise((resolve, reject) => {
  if(!process.env.REDIS_URL){
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  }

  const options = {
    type: 'redis',
    sentinels: [
      {
        host: 'redis_master',
        port: process.env.REDIS_SENTINEL_PORT
      }
    ,
      {
        host: 'redis_slave',
        port: process.env.REDIS_SENTINEL_PORT
      }
    ],
    password: process.env.REDIS_PASSWORD,
    family: 4,
    name: process.env.REDIS_REPLICATION_NAME,
    parser: 'javascript',
    dropBufferSupport: false,
    enableReadyCheck: true,
    connectTimeout: 10 * 1000,
    autoResubscribe: true,
    autoResendUnfulfilledCommands: true
  }

  const client = new Redis(options);

  client.on('error', function(err){
    console.error('Redis connect failed', err);
    reject(err);
  });

  client.on('ready', function(){
    console.log('Redis connected: %s', process.env.REDIS_URL);
    resolve();
  });
});

redpajama_set_name: RedPajamaGithub


applications for the 2018-2019 program year.
Your child must be at least three (3) years old by September 1, 2018. Please stop by one of our centers for additional information and to apply.
Head Start is a comprehensive child development program for low-income families. Ten percent (10%) of slots are reserved for children with disabilities.
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FCAA's Family Services Programs are designed to help families achieve independence. The comprehensive early childhood development program serves children from birth to age 5, pregnant women and their families.


We endeavor to consistently implement responsible corporate governance – taking into account both the expected returns of our shareholders and the long-term development of the Group. This represents a clear value structure in our global Group and supports our future outlook.
Consistent observance of the applicable legal regulations as well as consideration of the values and principles of the Group's own Code of Conduct, which we developed together with employees, form the basis of corporate action which is equally accepted by employees, customers and society.
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As a Luxembourg company, we have voluntarily allowed our compliance with the German Corporate Governance Code to be examined and presented.


text:
Haplosomoides serena is een keversoort uit de familie bladkevers (Chrysomelidae). De wetenschappelijke naam van de soort werd in 1859 gepubliceerd door Carl Henrik Boheman.

serena

redpajama_set_name: RedPajamaWikipedia


text:
Arsène Marie Joseph Edouard Gribomont (Bastenaken, 24 oktober 1878 - 8 april 1956) was een Belgisch senator.

Levensloop
Gribomont was de oudste van de tien kinderen van Edouard Gribomont (1850-1923) en van Jeanne Dujardin (1858-1947). Hij promoveerde tot doctor in de rechten en werd advocaat-pleitbezorger.

Van 1921 tot 1946 was hij provincieraadslid voor het kanton Bastenaken en vanaf 1934 was hij bestendig afgevaardigde voor de provincie Luxemburg.

In 1934 werd hij lid van de algemene vergadering van de Katholieke Unie van België als afgevaardigde van het arrondissementsverbond Bastenaken.

In 1946 werd hij verkozen tot PSC-senator voor het arrondissement Aarlen-Marche-Bastenaken, een mandaat dat hij vervulde tot aan zijn dood.

Literatuur
 Paul VAN MOLLE, Het Belgisch Parlement, 1894-1972, Antwerpen, 1972.

Externe link
 Fiche Gribomont op Odis

Belgisch advocaat
Belgisch senator
Gedeputeerde van Luxemburg
PSC-politicus


Wonder if Louisa is more popular for boys or girls?
This chart illustrates how many Girls were named Louisa in the U.S. since 1880.
Records indicate that 12,389 girls in the United States have been named Louisa since 1880.
The greatest number of people were given this name in 1916, when 225 people in the U.S. were given the name Louisa. Those people are now 102 years old.
What do you think about the name Louisa?


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Everyone desires a beautiful and unique look for their home garden. For their landscape, they want an appearance that no visitors will ever forget and that their neighbors will want to imitate. If you are looking for that stunning new, unique look for your Maryland home then it is time for you to consider creating a French garden for your landscape design. By focusing on symmetry and order, a French garden creates a formal look for your landscape that focuses on a centerpiece for the garden and leaves an incredible impression on anyone who sees it. Here are a few helpful tips on how to get started on creating a French garden for your home.
The basics of a French garden are simple to follow and play a big role in crafting a French garden. This means focusing on a cool color palette, focusing on white, blue, green, and purple colors in your French garden. You can easily do this by adding plants that can help fill in your borders. Keep in mind too to keep any hedges and shrubs neatly trimmed, stucco walls, and to include plenty of stonework in your French garden.
An important aspect of crafting a French garden is the amount of focus that is placed on stonework. Stone plays a large part in any French garden as it is used for paving, edgings, and also decorating your yard to add more to its overall image. Stonework is also great for a French garden because they can help create order within the garden by creating clear borders. Additionally, stone can be used in your French garden to help draw attention to special features, which is especially nice for your French garden's centerpiece.
Part of the appeal of a French garden is the emphasis of having an outdoor space that you want to spend time in, similar to the French countryside. This means having an area that you feel comfortable relaxing in, and even being able to eat in. This can easily be achieved by adding a few pieces of simple furniture and including an outdoor kitchen to help encourage a desire to be outside and eat.
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Home/Celebrities/Bobbi Kristina Brown Update – Family Says She's 'Fighting For Her Life'
Bobbi Kristina Brown Update – Family Says She's 'Fighting For Her Life'
Netloid Team February 2, 2015
Two days after Bobbi Kristina Brown was hospitalized, her family has released a statement, saying she's "fighting for her life."
The daughter of Bobby Brown and the late Whitney Houston was found unresponsive in her Roswell, Georgia, home on Saturday and taken to North Fulton Hospital, where she was reportedly placed in a medically induced coma. According to People, she is breathing with the help of a ventilator.
In a statement released Monday, the 21-year-old's family said, "Bobbi Kristina is fighting for her life and is surrounded by immediate family. As her father already stated, we are asking you to honor our request for privacy during this difficult time. Thank you for your prayers, well wishes, and we greatly appreciate your continued support."
Bobby Brown previously released a statement on Sunday, which read, "Privacy is requested in this matter. Please allow for my family to deal with this matter and give my daughter the love and support she needs at this time."
Although a police report states that officers responded in "reference [to] a drowning" at the home of Bobbi Kristina, the circumstances by which she came to be unresponsive remain under investigation. According to Lisa Holland of the Roswell Police Department, "There were nothing there that indicated that anything out of the ordinary happened. There were no drugs out in the open."
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Prices up but value growth 'very modest,' says QV
Landlords.co.nz
Nationwide values are up 1.1% over the past three months, 2.9% up over the past year and are now 2.9% below the previous market peak of late 2007.
QV research director Jonno Ingerson said however that the increases need to be viewed in light of past price rises.
"Although national values have increased 2.9% over the past year this needs to be compared to previous periods of value growth," he said.
"During 1993 to 1997 values increased between 8% and 14% per year. Then in the 2002 to 2007 boom values generally increased by 10% to 15% per year, and increased by a staggering 25% during 2003 alone.
"In comparison the current rate of value increase is very modest and in inflation adjusted terms is only just above level."
Ingerson said the market had seen an increase in activity over the last month which is typical for the time of year, but while higher than the past few years activity still remains below long term averages.
"There is still a shortage of properties for sale in some areas, and in general buyers are acting cautiously and carefully."
QV data revealed Auckland remains the fastest growing of the main centres, up 1.7% in the past three months and 4.8% up on last year, with values now 2.3% above the last market peak.
The Auckland market increases are being led by the old Auckland City, where values are up 6.5% over the year to 5.1% above the market peak of 2007.
"There appears to be strong demand in the central areas of old Auckland City such as Ponsonby, Grey Lynn and Sandringham. Healthy sales prices are being achieved, with good properties coming to the market. Some buyers are staying cautious and sticking to their upper limit whereas others seem to have few financial constraints and as a result auction prices are setting new levels" said QV Valuer Glenda Whitehead.
"Buyers in the North Shore are being driven by location, as opposed to price, with activity patchy. Some suburbs are experiencing good turnover of properties whilst others are quite stagnant. The number of private sales is also noticeable."
"In the West more renovations are noticeable, with people wanting to upgrade or extend their homes to meet their needs as opposed to incurring buying and selling costs. In comparison, the market in the Manukau area is active around the $500,000 - $800,000 range, even up to $1 million. Houses in the typically desirable areas such as old Howick properties with sea views, and houses in sought after school zones are experiencing good sales," she said.
In the capital values have remained at the same level as a year ago after dropping 2% for the first six months, then steadily rising for the last six months.
"Across Wellington, prices seem to be firming and more listings are coming to the market. Buyers are still being cautious, especially with the public sector restructuring and its effect on job security in the region" said QV Valuer Pieter Geill.
Values in Hamilton remain relatively stable, rising 0.9% over the past year, but are currently 11.1% below the 2007 peak.
"The number of sales appears to be the highest for five years, with homes in the mid to upper bracket appearing to be selling well especially. This renewed interest however, is not stimulating prices with values remaining consistently static" said QV Valuer Richard Allen.
Tauranga values have also remained relatively static, up 1% over the past year but remaining 11.3% below the 2007 peak.
QV Valuer Shayne Donovan-Grammer said, "Many homes, particularly in the better regarded suburbs, are slightly overpriced in this market. Buyers are still looking to pay a sharp price, which is reflected in the fact that realistically priced well presented homes in the low to mid value range are selling at a good rate."
Apart from Auckland City values in Christchurch have grown faster over the past year than any of the other main centres, rising 4% and are now level with the 2007 market peak.
Waimakariri District has increased 12% over the past year and Selwyn District 10.8%. Both areas are by far the fastest increasing parts of the country. They are also the highest about the 2007 market peak at 6.5% and 5.9% respectively.
QV Valuer Richard Kolff said, "Rents continue to rise with high demand placed on rental properties from people whose homes were uninhabitable due to earthquake damage, people who need to relocate whilst they carry out repairs, or from those who are moving to the city to help with the rebuild. This is making property investors look at what potential some properties now have."
Dunedin values are 2.3% above the same time last year, although the growth of 4% over the past six months is even faster than Auckland City.
"The lower end of the market is still seeing reasonable demand with purchasers now prepared to take on more risk with lower priced houses. This is in contrast to six months ago when most of the demand was for well presented dwellings with no deferred maintenance issues" said QV Valuer Tim Gibson.
Values in most of the provincial centres have stayed relatively stable for the past three months, with the exception of Gisborne (-3.5%) and Wanganui (-2.0%). Over the past year values are within 2% of last year's value in all centres apart from Napier (-2.2%), Wanganui (-5.2%) and Queenstown Lakes ( 2.7%).
Tags: tim gibson - jonno ingerson - glenda whitehead - pieter geill - shayne donovan-grammer - richard kolff
Source: Landlords.co.nz
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The sewn out size for the 2.40" x 1.60" hoop is 1.33" W x 1.31" H with 1,786 stitches.
The sewn out size for the 4" x 4" hoop is 2.74" W x 2.70" H with 4,751 stitches.
The sewn out size for the 5" x 7" hoop is 4.57" W x 4.51" H with 13,159 stitches.
The sewn out size for the 6.25" x 10.25" hoop is 5.94" W x 5.87" H with 22,349 stitches.
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The Chester Beatty Library is an art museum and library which houses a priceless collection of early Christian and Islamic Manuscripts from the Middle East the museum's collection also comprises miniature paintings, prints, drawings, rare books and some decorative arts assembled by Sir Alfred Chester Beatty (1875-1968). Egyptian papyrus texts, beautifully illuminated copies of the Qur'an, the Bible, European medieval and renaissance manuscripts are among the highlights of the collection.
Dr David Hutchinson Edgar (below, second from left) author of "Treasuring the Word: An Introduction to Biblical Manuscripts in the Chester Beatty Library ", kindly acted as guide to our guests.
Phillip O'Connor; David Morrison; Alan Lonergan; Seham Musallam; Peter Byrne.
The viewing was followed by lunch at The Silk Road Café in the Library.
Each of our guests addressed the Committee and answered questions from Committee members. The transcript of the meeting is available from the Oireachtas here.
"I would like to make a brief comment on behalf of Sadaka. I would like to express concern over Deputy Barrett's comment that the international community and the negotiations need to be approached with balance. This is not a balanced situation. This is not a conflict of equals. This is an occupation. We have an occupier and we have an occupied people. We have an oppressor and we have an oppressed people. We have a powerful people and we have a powerless people. To approach that situation with balance is to side with the occupier. The international community does not need to approach the situation with a sense of balance, but needs to exert pressure on the state of Israel until such time as it complies with international law and ends the occupation."
The Lord Mayor of Dublin, Cllr. Gerry Breen hosted a reception for the visiting church delegation in the Mansion House, Dublin. Invited guests included government, trade union and church leaders as well as supporters of Sadaka and representatives of NGOs active on Palestine.


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Charles Cartin, Ph.D., named inaugural director of makerspaces at VCU Engineering. [View Image]
Charles Cartin, Ph.D., named inaugural director of makerspaces at VCU Engineering.
VCU Engineering appoints new director of makerspaces
Hands-on learning is central to VCU College of Engineering's philosophy. As the college dedicates more square footage to active, collaborative design and fabrication, it has named Charles Cartin, Ph.D., as its inaugural director of makerspaces. Cartin is an associate professor in the Department of Mechanical and Nuclear Engineering.
"Makerspaces are places where students from all walks of life can come together to innovate, ideate, create and build something together from start to finish," Cartin said. "Allowing engineers, art majors and students from other fields to unite on multiple projects is a way of opening the door for extended collaborations across all departments."
He will continue to lead the MNE Innovation Lab. Established in 2016, this advanced additive manufacturing hub offers an array of 3D printers for large and small production. To date, it has supported the work of faculty and students from disciplines including dentistry, medicine, the arts, as well as engineering.
In his new role, Cartin will also oversee the operations of VCU's Maker Garage, slated to open in September. This newly renovated, 4,000-square-foot facility at 12 W. Cary Street, formerly the site of Grubbs' Auto Service, has traditional and advanced manufacturing capabilities, including resources for large-scale projects such as Hyperloop and Formula SAE at VCU.
The Maker Garage has spaces for metalworking, assembly and finishing, plus a computing and conference area so teams can modify their plans collaboratively. Like all of the college's makerspaces, it supports interdisciplinary thinking, so it will be used — and staffed by — students from multiple academic fields. Boris Solomonov, the college's machine shop manager, will assist with student projects, training and workflow.
When VCU's new Engineering Research Building opens in 2020, Cartin will also direct the 9,000-square-foot Maker Studio that will occupy much of the building's ground floor. Plans for this expansive facility are underway and will include spaces for large and small group collaboration, storage for long-term projects and a variety of old-school and next-generation manufacturing tools.
These facilities' equipment — which includes laser cutters, computer numerical control machines and the Maker Garage's new waterjet cutter capable of safely cutting almost any material — is dazzling, but Cartin thinks active problem solving will be the most enduring benefit.
"Giving students a hands-on approach to the process allows them to actually realize their design work well before they finalize the overall product," he said. "This makes designs and the designers better."