Handling the crossing minimization problem

1. Why the crossing minimization problem is so hard...
2. A general approximation result for n+cr(G).
3. The planarization heuristic, and planar insertion problems.
4. A surprising hardness result about inserted edge.
5. Crossing approximation algorithms based on insertion solutions.

Advanced algorithmic techniques

1. Structural and topological upper bounds on the crossing number.
   
2. Constant-factor approximation algorithms for embedded graphs.
3. Exact ILP approach to crossing minimization.