- To what degree does the number of unconnected k-hop solutions which have at most as many nodes as an optimal connected solution affect the runtime? (Calculate how many of them exist and add this data as a column to a table of test results)

- To what degree does the number of unconnected k-hop solutions which have at most as many nodes as an optimal connected solution affect the runtime? (Calculate how many of them exist and add this data as a column to a table of test results)

- Is there a clear corellation between the number of constraints which were added and the runtime for (possible unconnected) solutions?

- Is there a clear corellation between the number of constraints which were added and the runtime for (possible unconnected) solutions?

- Check for graphs where the gap between an optimal connected solution and the smallest unconnected solution is rather big if this approach(our implementation) might be unapplyable. Such that there is too much time wasted to exclude all possible unconnected solutions and increase the lower bound.

- Check for graphs where the gap between an optimal connected solution and the smallest unconnected solution is rather big if this approach(our implementation) might be unapplyable. Such that there is too much time wasted to exclude all possible unconnected solutions and increase the lower bound.

* Implement a heuristic to start with a stronger lower bound.

* Implement a heuristic to start with a stronger lower bound. ✔

- Given a symmetrical leaf where the root is centered (at least on the x-axis). There are at least ![width-2*k+height-k-\frac{width}{2}-k](https://render.githubusercontent.com/render/math?math=width-2*k%2Bheight-k-%5Cfrac%7Bwidth%7D%7B2%7D-k) nodes needed to provide a connected dominating set.

- Given a symmetrical leaf where the root is centered (at least on the x-axis). There are at least ![width-2*k+height-k-\frac{width}{2}-k](https://render.githubusercontent.com/render/math?math=width-2*k%2Bheight-k-%5Cfrac%7Bwidth%7D%7B2%7D-k) nodes needed to provide a connected dominating set.

- To reach both rims in the width and the rim on the top.

- To reach both rims in the width and the rim on the top.

- Unfortunately this did not improve the runtime at least for the bigger-leaf example with k=2. It drastically increased the runtime.

- So at first I would not follow this idea and try to find another technique to generate a better lower bound.

## thesis

## thesis

* Begin paragraph about results and add preliminary results and observations.

* Begin paragraph about results and add preliminary results and observations.

...

@@ -60,7 +62,7 @@

...

@@ -60,7 +62,7 @@

# Backlog

# Backlog

## Python

## Python

### Implementation

### Implementation

* Remove lp_to_graph from package and instead use some other format to store graphs and use them as input.

* Remove lp_to_graph from package and instead use some other format to store graphs and use them as input. ✔

* Implement different type of constraints.

* Implement different type of constraints.

- _Solving the Maximum-Weight Connected Subgraph Problem to Optimality_

- _Solving the Maximum-Weight Connected Subgraph Problem to Optimality_