Planar Multiway Cut with Terminals on Few Faces

TL;DR

This paper presents a new algorithm for the Edge Multiway Cut problem on planar graphs, achieving an $n^{O(\sqrt{k})}$ time complexity where $k$ is the number of faces covering all terminals, improving understanding of parameterized complexity.

Contribution

The authors introduce a novel approach combining homotopy, sphere-cut decomposition, and dynamic programming to solve Edge Multiway Cut efficiently based on face coverage.

Findings

Achieves $n^{O(\sqrt{k})}$ time algorithm for Edge Multiway Cut with face parameter.

Employs homotopy and sphere-cut decomposition techniques.

Combines global treewidth and local Dreyfus-Wagner style dynamic programming.

Abstract

We consider the \textsc{Edge Multiway Cut} problem on planar graphs. It is known that this can be solved in $n^{O(\sqrt{t})}$ time [Klein, Marx, ICALP 2012] and not in $n^{o(\sqrt{t})}$ time under the Exponential Time Hypothesis [Marx, ICALP 2012], where $t$ is the number of terminals. A stronger parameter is the number $k$ of faces of the planar graph that jointly cover all terminals. For the related {\sc Steiner Tree} problem, an $n^{O(\sqrt{k})}$ time algorithm was recently shown [Kisfaludi-Bak et al., SODA 2019]. By a completely different approach, we prove in this paper that \textsc{Edge Multiway Cut} can be solved in $n^{O(\sqrt{k})}$ 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.