What is the LP gap of vertex cover in planar graphs?

The LP I refer to is min $\sum_{e \in E } c_e x_e \ \ $ subject to $ \ \ x_v + x_u \geq 1 \ \ \ \forall uv \in E $

$ c_e \geq 0 $ are costs of the vertices

Also is there an LP with a smaller gap?

(I posted this in mathstackexchange where it was labelled offtopic if it is offtopic here please tell me where I should post this.)