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 # Solving the Degree Constrained Bottleneck Spanning Tree Problem with CP-SAT
+
+## Problem Definition
+
+## Selection of Variables for the Model
+
+The most trivial variable selection would be to choose a boolean variable for every edge, that represents if the edge is in the solution.
+Unfortunately, with this representation it is difficult to efficiently contrain the edge set to be a tree.
+Instead we will work on an [arborescence](https://en.wikipedia.org/wiki/Arborescence_(graph_theory)) (instead of an undirected tree) and keep track of how deep a vertex is in this arborescence.
+You will often need to create auxiliary variables or constructs that allow you to enforce your constraints more efficiently.
+After some time, you acquire a set of such techniques that allow you to model nearly anything without much thinking.
+In the beginning, a lot may seem unelegant or random.
+
+* Arc variables $x_{vw}\in \{0,1\}$ and $x_{wv}\in \{0,1\}$ for every edge $\{v, w\} \in E$ that represent with $x_{vw}=1$ if the arc is used for the arborescence.
+    * An edge $\{v,w\}$ will be in the solution if either $x_{vw}=1$ or $x_{wv}=1$.
+* Bottleneck variable $y\in \{0,1,\ldots,\max_{\{v,w\}in E}(d(v,w))\}$ that represent the weight of the most expensive used edge/arc.
+* Depth variables $d_v \in \{0, 1, \ldots, |V|-1\}$ for every vertex $v\in V$ representing how deep the vertex is in the arborescence.
+
+## Objective Function
+
+The objective function is in this case trivial as we created the auxiliary variable $y$.
+As long as we ensure that $y$ actually gets assigned the value of the most expensive selected edge, we just have to specify
+$$\min y$
+to obtain the feasible solution with the smallest possible $y$-value.
+
+## Constraints
+
+We have specified what a solution looks like and how to measure its quality.
+Next, we have to make sure it actually obeys the rules as otherwise we just get the trivial zero solution that is of perfect zero-weight, but unfortunately not a feasible solution.
+
+Let us start with some simple constraints that don't need much explanation:
+
+* Only one direction of an edge can be chosen: $\forall \{v,w\}\in E: x_{vw}+x_{wv}\leq 1$
+* We need to select $|V|-1$ edges for a tree: $\sum_{\{v,w\}\in E}x_{vw}+x_{wv} = |V|-1$
+* 
+
+### y-Variable represents most expensive selected edge.
+
+An edge $\{v,w\}$ is selected if $x_{vw}=1$ or $x_{wv}=1$.
+Thus, if $x_{vw}=1$, $y\geq d(v,w)$.
+$$y \geq d(v,w) \quad \text{if }x_{vw}=1 \quad \forall vw \text{ with } \{v, w\}\in E$$
+The objective will make sure that it is not larger than necessary, setting it equal to the most expensive selected edge.
+
+
+### Tree constraint
+
+A tree needs to have $|V|-1$ edges, so we enforce
+$$ \sum_{\{v, w\}\in E} x_{vw}+x_{wv} = |V|-1$$
+
+To make sure it does not have cycles, 
+
+### Degree constraint
+
+