Edge disjoint paths and max integral multiflow/min multicut theorems in planar graphs

We generalize all the results obtained for integer multiflow and multicut problems in trees by Garg et al. [N. Garg, V.V. Vazirani and M. Yannakakis, Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica 18 (1997) 3–20] to planar graphs with a fixed number of faces, although other classical generalizations do not lead to such results. We also introduce the class of k-edge-outerplanar graphs and bound the integrality gap for the maximum edge-disjoint paths problem in these graphs.