A 1.5-pproximation algorithms for activating 2 disjoint $st$-paths

TL;DR

This paper investigates the Activation 2-Disjoint $st$-Paths problem, correcting a previous claim of a 1.5-approximation algorithm and providing a new algorithm with the claimed ratio.

Contribution

The paper corrects the error in the previous 1.5-approximation claim and introduces a new algorithm achieving a 1.5-approximation ratio for the problem.

Findings

The previous proof claiming a 1.5-approximation is incorrect.

The authors establish that the approximation ratio of the previous algorithm is at least 2.

A new algorithm with a proven 1.5-approximation ratio is presented.

Abstract

In the $Activation$ $k$ $Disjoint$ $st$-$Paths$ ($Activation$ $k$-$DP$) problem we are given a graph $G=(V,E)$ with activation costs $\{c_{uv}^u,c_{uv}^v\}$ for every edge $uv \in E$, a source-sink pair $s,t \in V$, and an integer $k$. The goal is to compute an edge set $F \subseteq E$ of $k$ internally node disjoint $st$-paths of minimum activation cost $\displaystyle \sum_{v \in V}\max_{uv \in E}c_{uv}^v$. The problem admits an easy $2$-approximation algorithm. Alqahtani and Erlebach [CIAC, pages 1-12, 2013] claimed that Activation 2-DP admits a $1.5$-approximation algorithm. Their proof has an error, and we will show that the approximation ratio of their algorithm is at least $2$. We will then give a different algorithm with approximation ratio $1.5$.