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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 nO(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].

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