A Tight Lower Bound for Planar Multiway Cut with Fixed Number of Terminals

Abstract

Given a planar graph with k terminal vertices, the Planar Multiway Cut problem asks for a minimum set of edges whose removal pairwise separates the terminals from each other. A classical algorithm of Dahlhaus et al. [2] solves the problem in time n ^O(k), which was very recently improved to $2^{O(k)}\cdot n^{O(\sqrt{k})}$ time by Klein and Marx [6]. Here we show the optimality of the latter algorithm: assuming the Exponential Time Hypothesis (ETH), there is no $f(k)\cdot n^{o(\sqrt{k})}$ time algorithm for Planar Multiway Cut. It also follows that the problem is W[1]-hard, answering an open question of Downey and Fellows [3].