Shortest two disjoint paths in conservative graphs

TL;DR

This paper introduces a polynomial-time algorithm for finding two disjoint shortest paths in undirected graphs with conservative edge weights, specifically when negative edges form a limited number of trees, extending the class of tractable instances.

Contribution

The paper presents the first polynomial-time solution for the Shortest Two Disjoint Paths problem with conservative weights under the condition that negative edges form a constant number of trees.

Findings

The algorithm efficiently solves the problem for graphs with a limited number of negative-weight trees.

It extends the class of graphs where the problem is tractable despite negative weights.

The problem remains NP-hard in general, but is solvable in polynomial time under the specified conditions.

Abstract

We consider the following problem that we call the Shortest Two Disjoint Paths problem: given an undirected graph $G=(V,E)$ with edge weights $w:E \rightarrow \mathbb{R}$, two terminals $s$ and $t$ in $G$, find two internally vertex-disjoint paths between $s$ and $t$ with minimum total weight. As shown recently by Schlotter and Seb\H{o} (2022), this problem becomes NP-hard if edges can have negative weights, even if the weight function is conservative, there are no cycles in $G$ with negative total weight. We propose a polynomial-time algorithm that solves the Shortest Two Disjoint Paths problem for conservative weights in the case when the negative-weight edges form a constant number of trees in $G$.