Using a geometric lens to find k disjoint shortest paths

TL;DR

This paper introduces a geometric approach to solve the k-Disjoint Shortest Paths problem more efficiently and demonstrates its limitations through complexity hardness results.

Contribution

It presents an improved algorithm with a novel geometric perspective and refines the special case for k=2, while also proving W[1]-hardness for the problem.

Findings

New geometric algorithm with $n^{O(k!k)}$ complexity

Special case $k=2$ solved in $O(nm)$ time

Proved W[1]-hardness with respect to $k$

Abstract

Given an undirected $n$-vertex graph and $k$ pairs of terminal vertices $(s_1,t_1), \ldots, (s_k,t_k)$, the $k$-Disjoint Shortest Paths ($k$-DSP)-problem asks whether there are $k$ pairwise vertex-disjoint paths $P_1,\ldots, P_k$ such that $P_i$ is a shortest $s_i$-$t_i$-path for each $i \in [k]$. Recently, Lochet [SODA 2021] provided an algorithm that solves $k$-DSP in $n^{O(k^{5^k})}$ time, answering a 20-year old question about the computational complexity of $k$-DSP for constant $k$. On the one hand, we present an improved $n^{O(k!k)}$-time algorithm based on a novel geometric view on this problem. For the special case $k=2$ on $m$-edge graphs, we show that the running time can be further reduced to $O(nm)$ by small modifications of the algorithm and a refined analysis. On the other hand, we show that $k$-DSP is W[1]-hard with respect to $k$, showing that the dependency of the degree of the polynomial running time on the parameter $k$ is presumably unavoidable.