It would be fun to have a list of misleading ideas and traps in CO.

1 Polytope for s-t cut

$$\begin{array}{c}\begin{array}{rlrl}\sum _{e\in p}{x}_e& \ge 1& & \forall \text{ s-t path p}\\ x_e& \ge 0& & \forall e\in E\end{array}\end{array}$$ Is this polytope integral?

Initially I thought, we have max flow min cut thm and flow polytope is integral and this path formulation of s-t cut should be the dual of flow problem and thus it is integral. However, hitting spanning tree (dual to tree packing) formulation has a integrality gap of 2. Make sense.

However, LP(1) is not the dual of Max-flow.

2 Size of support

Given a linear program with a rank $r$ constraint matrix, what is the size of support of its optimal solution?

This is mentioned in a previous post. The description there is not precise. Consider