Parameterized Algorithm for the Disjoint Path Problem on Planar Graphs: Exponential in k² and Linear in n

In this paper, we study the Planar Disjoint Paths problem: Given an undirected planar graph G with n vertices and a set T of k pairs (si, ti)ki=1 of vertices, the goal is to find a set P of k pairwise vertex-disjoint paths connecting si and ti for all indices i ∈ {1,…, k}. We present a 2O(k2)n-time algorithm for the Planar Disjoint Paths problem. This improves the two previously best-known algorithms: 22O(k)-time algorithm [Discrete Applied Mathematics 1995] and 2O(k2)n6-time algorithm [STOC 2020].* The full version of the paper can be accessed at https://arxiv.org/abs/2211.03341† This work was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No.2020R1C1C1012742).