Planar Disjoint Shortest Paths is Fixed-Parameter Tractable

In the Disjoint Shortest Paths problem one is given a graph $G$ and a set $\mathcal{T} = \{(s_1,t_1),\ldots,(s_k,t_k)\}$ of $k$ vertex pairs. The question is whether there exist vertex-disjoint paths $P_1,\ldots,P_k$ in $G$ so that each $P_i$ is a shortest path between $s_i$ and $t_i$. While the problem is known to be $\textsf{W}$[1]-hard in general, we show that it is fixed-parameter tractable on planar graphs with positive edge weights. Specifically, we propose an algorithm for Planar Disjoint Shortest Paths with running time $2^{\mathcal{O}(k \log k)} \cdot n^{\mathcal{O}(1)}$. Notably, our parameter dependency is better than state-of-the-art $2^{\mathcal{O}(k^2)}$ for the Planar Disjoint Paths problem, where the sought paths are not required to be shortest paths.