The hardness of routing two pairs on one face

TL;DR

This paper proves that finding disjoint paths connecting two pairs on a face in a planar graph is NP-complete, resolving open problems and extending complexity results to directed acyclic graphs with minimal demand edges.

Contribution

It establishes the NP-completeness of the two-pair routing problem on a face in planar graphs and extends this complexity to directed acyclic graphs with only two demand arcs.

Findings

NP-completeness of two-pair routing on a face in planar graphs

NP-completeness of arc-disjoint paths in directed acyclic graphs with two demand arcs

Strengthens previous complexity results in planar graph disjoint path problems

Abstract

We prove the NP-completeness of the integer multiflow problem in planar graphs, with the following restrictions: there are only two 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 M\"uller. It also strengthens Schw\"arzler's recent proof of one of the open problems of Schrijver's book, 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 demand arcs.