Planar Multiway Cut with Terminals on Few Faces

We consider the Edge Multiway Cut problem on planar graphs. It is known that this can be solved in n O( √ t) time [Klein, Marx, ICALP 2012] and not in n o( √ t) time under the Exponential Time Hypothesis [Marx, ICALP 2012], where t is the number of terminals. A generalization of this parameter is the number k of faces of the planar graph that jointly cover all terminals. For the related Steiner Tree problem, an n O( √ k)

time algorithm was recently shown [Kisfaludi-Bak et al., SODA 2019]. By a completely different approach, we prove in this paper that Edge Multiway Cut can be solved in n O( √ k) time as well. Our approach employs several major concepts on planar graphs, including homotopy and sphere-cut decomposition. We also mix a global treewidth dynamic program with a Dreyfus-Wagner style dynamic program to locally deal with large numbers of terminals.