Many graph problems that are hard to approximate on general graphs become much more tractable on
planar graphs. In particular, planar graphs can be decomposed into small pieces or into bounded
treewidth graphs, leading to PTASes for these problems. But little is known about the noisy setting
where the graphs are only nearly planar, i.e. deleting few edges makes them planar. 
One obstacle is that current planar decomposition techniques fail completely with noise. 
Another obstacle is that the known guarantees for the planarization problem are too weak for our purpose.

We show that using linear programming methods such as configuration LPs and spreading metrics, one can get 
around these barriers and obtain PTASes for many problems on noisy planar graphs. 
This resolves an open question of Magen and Moharrami, that was recently popularized by Claire Mathieu.