  In this paper, we bound the integrality gap and the approximation ratio for
maximum plane multiflow problems and deduce bounds on the flow-cut-gap.
Planarity means here that the union of the supply and demand graph is planar.
We first prove that there exists a multiflow of value at least half of the
capacity of a minimum multicut. We then show how to convert any multiflow into
a half-integer one of value at least half of the original multiflow. Finally,
we round any half-integer multiflow into an integer multiflow, losing again at
most half of the value, in polynomial time, achieving a $1/4$-approximation
algorithm for maximum integer multiflows in the plane, and an integer-flow-cut
gap of $8$.
