The planar directed k-Vertex-Disjoint Paths problem is fixed-parameter tractable

TL;DR

This paper proves that the planar directed k-Vertex-Disjoint Paths problem is fixed-parameter tractable by providing an algorithm with a double-exponential dependence on k and polynomial dependence on the input size.

Contribution

The authors present a new fixed-parameter algorithm for the planar directed k-Vertex-Disjoint Paths problem, improving upon previous complexity bounds.

Findings

The problem is fixed-parameter tractable with respect to k.

The algorithm runs in 2^{2^{O(k^2)}} n^{O(1)} time.

This advances understanding of disjoint paths in planar directed graphs.

Abstract

Given a graph G and k pairs of vertices (s_1,t_1), ..., (s_k,t_k), the k-Vertex-Disjoint Paths problem asks for pairwise vertex-disjoint paths P_1, ..., P_k such that P_i goes from s_i to t_i. Schrijver [SICOMP'94] proved that the k-Vertex-Disjoint Paths problem on planar directed graphs can be solved in time n^{O(k)}. We give an algorithm with running time 2^{2^{O(k^2)}} n^{O(1)} for the problem, that is, we show the fixed-parameter tractability of the problem.