Edge-Disjoint Paths in Planar Graphs with Short Total Length

The problem of nding edge{disjoint paths in a planar graph such that each path connects two speciied vertices on the outer face of the graph is well studied. The \classical" Eulerian case introduced by Okamura and Seymour 7] is solvable in linear time 10]. So far, the length of the paths were not considered. In this paper now, we prove that the problem of nding edge{disjoint paths of minimum total length in a planar graph is N P{hard, even if the graph fullllls the Eulerian condition and the maximum degree is four. Minimizing the length of the longest path is N P{hard as well. EEcient heuristics based on the algorithm from 10] are presented that determine edge{disjoint paths of small total length. We have implemented these heuristics and have studied their behaviour. It turns out that some of the heuristics are empirically very successful.