Parameterized Algorithm for the Disjoint Path Problem on Planar Graphs: Exponential in $k^2$ and Linear in $n$

TL;DR

This paper introduces a new parameterized algorithm for the Planar Disjoint Paths problem that runs in exponential time in $k^2$ but linear in the number of vertices, improving upon previous algorithms.

Contribution

The paper presents a more efficient fixed-parameter algorithm for the Planar Disjoint Paths problem with a runtime of $2^{O(k^2)}n$, surpassing earlier methods.

Findings

Achieved a $2^{O(k^2)}n$-time algorithm for the problem.

Improved the computational complexity over previous algorithms.

Demonstrated the algorithm's efficiency for large planar graphs.

Abstract

In this paper, we study the \textsf{Planar Disjoint Paths} problem: Given an undirected planar graph $G$ with $n$ vertices and a set $T$ of $k$ pairs $(s_i,t_i)_{i=1}^k$ of vertices, the goal is to find a set $\mathcal P$ of $k$ pairwise vertex-disjoint paths connecting $s_i$ and $t_i$ for all indices $i\in\{1,\ldots,k\}$. We present a $2^{O(k^2)}n$-time algorithm for the \textsf{Planar Disjoint Paths} problem. This improves the two previously best-known algorithms: $2^{2^{O(k)}}n$-time algorithm [Discrete Applied Mathematics 1995] and $2^{O(k^2)}n^6$-time algorithm [STOC 2020].