In the NP-hard *Degree-Constrained Bottleneck Spanning Tree (DBST)*, we are given a complete graph $G=(V,E)$ with distance/weight function $dist: E \rightarrow \mathbb{N}^+_0$.
In the NP-hard *Degree-Constrained Bottleneck Spanning Tree (DBST)*,
we are given a complete graph $G=(V,E)$ with distance/weight function $dist: E \rightarrow \mathbb{N}^+_0$.
The set $V$ can be a set of point on the plane, and $dist(v,w)$ the euclidean distance between $v\in V$ and $w\in V$.
Additionally, we have a degree constraint $d\geq 2$.
The objective is to find a tree $T$ in $G$ in which no vertex has a degree greater the degree constraint, i.e., $\forall v\in V: deg_T(v)\leq d$, and the weight of the longest edge is minimized.
## Modelling the Problem with CP-SAT
Here we model the problem declerative to be solved with CP-SAT.
The code can be found in [_solver.py](./_solver.py).
### Selection of Variables for the Model
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@@ -31,7 +33,7 @@ See `__make_vars` in the code.
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
$$\min y$
$$\min y$$
to obtain the feasible solution with the smallest possible $y$-value.
### Constraints
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@@ -42,12 +44,12 @@ Next, we have to make sure it actually obeys the rules as otherwise we just get
Let us start with some simple constraints that don't need much explanation:
* Only one direction of an edge can be chosen: $\forall \{v,w\}\in E: x_{vw}+x_{wv}\leq 1$ (see `__forbid_bidirectional_edges`)
* We set a random vertex $v_0$ as root and enforce that every other vertex has a parent.
* We set a random vertex $v_0$ as root and enforce that every other vertex have a parent. (See `__add_degree_constraints`)
