A Linear Time Algorithm for Finding Three Edge-Disjoint Paths in Eulerian Networks

TL;DR

This paper presents a linear time combinatorial algorithm for finding three edge-disjoint paths connecting specified terminal pairs in Eulerian networks, improving upon previous polynomial algorithms.

Contribution

It introduces a new linear time algorithm specifically for Eulerian networks, leveraging cut conditions to efficiently find the paths.

Findings

The algorithm runs in O(m) time for Eulerian networks.

It guarantees construction of the paths if they exist.

It simplifies the problem by using Eulerian properties and cut conditions.

Abstract

Consider an undirected graph $G = (VG, EG)$ and a set of six \emph{terminals} $T = \set{s_1, s_2, s_3, t_1, t_2, t_3} \subseteq VG$. The goal is to find a collection $\calP$ of three edge-disjoint paths $P_1$, $P_2$, and $P_3$, where $P_i$ connects nodes $s_i$ and $t_i$ ($i = 1, 2, 3$). Results obtained by Robertson and Seymour by graph minor techniques imply a polynomial time solvability of this problem. The time bound of their algorithm is $O(m^3)$ (hereinafter we assume $n := \abs{VG}$, $m := \abs{EG}$, $n = O(m)$). In this paper we consider a special, \emph{Eulerian} case of $G$ and $T$. Namely, construct the \emph{demand graph} $H = (VG, \set{s_1t_1, s_2t_2, s_3t_3})$. The edges of $H$ correspond to the desired paths in $\calP$. In the Eulerian case the degrees of all nodes in the (multi-) graph $G + H$ ($ = (VG, EG \cup EH)$) are even. Schrijver showed that, under the assumption of Eulerianess, cut conditions provide a criterion for the existence of $\calP$. This, in particular, implies that checking for existence of $\calP$ can be done in $O(m)$ time. Our result is a combinatorial $O(m)$-time algorithm that constructs $\calP$ (if the latter exists).