Abstract
We prove the NP-completeness of the integer multiflow problem in planar graphs, with the following restrictions: there are only two classes of parallel demand edges, both lying on the infinite face of the routing graph. This was one of the open challenges concerning disjoint paths, explicitly asked by Muller (Math Program 105 (2–3):275–288, 2006). It also strengthens Schwarzler’s recent proof of one of the open problems of Schrijver’s book (Combinatorial Optimization: Polyhedra and Efficiency. Springer, Berlin 2003), about the complexity of the edge-disjoint paths problem with terminals on the outer boundary of a planar graph. We also give a directed acyclic reduction. This proves that the arc-disjoint paths problem is NP-complete in directed acyclic graphs, even with only two classes of demand arcs.
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