Vertex Disjoint Paths in Upward Planar Graphs

Abstract

The k-vertex disjoint paths problem is one of the most studied problems in algorithmic graph theory. In 1994, Schrijver proved that the problem can be solved in polynomial time for every fixed k when restricted to the class of planar digraphs and it was a long standing open question whether it is fixed-parameter tractable (with respect to parameter k) on this restricted class. Only recently, Cygan et al. [5] achieved a major breakthrough and answered the question positively. Despite the importance of this result, it is of rather theoretical importance. Their proof technique is both technically extremely involved and also has a doubly exponential parameter dependence. Thus, it seems unrealistic that the algorithm could actually be implemented. In this paper, therefore, we study a smaller but well studied class of planar digraphs, the class of upward planar digraphs which can be drawn in a plane such that all edges are drawn upwards. We show that on this class the problem (i) remains NP-complete and (ii) problem is fixed-parameter tractable. While membership in FPT follows immediately from [5]’s general result, our algorithm is very natural and has only singly exponential parameter dependence and linear dependence on the graph size, compared to the doubly exponential parameter dependence and much higher polynomial dependence on the graph size for general planar digraphs. Furthermore, our algorithm can easily be implemented, in contrast to the algorithm in [5].