Integer Plane Multiflow Maximisation: Flow-Cut Gap and One-Quarter-Approximation

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 $8$.