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 $1.5$.

翻译：在 $Activation$ $k$ $Disjoint$ $st$-$Paths$ ($Activation$ $k$-$DP$) 问题中，给定一个图 $G=(V,E)$，每条边 $uv \in E$ 具有激活成本 $\{c_{uv}^u,c_{uv}^v\}$，一个源-汇节点对 $s,t \in V$，以及一个整数 $k$。目标是计算一个边集 $F \subseteq E$，使其包含 $k$ 条内部节点不相交的 $st$-路，且最小化激活成本 $\displaystyle \sum_{v \in V}\max_{uv \in E}c_{uv}^v$。该问题存在一个简单的 $2$-近似算法。Alqahtani 和 Erlebach [CIAC, 页码 1-12, 2013] 声称 Activation 2-DP 存在一个 $1.5$-近似算法，但其证明存在错误，我们将指出该算法的近似比至少为 $2$。随后我们将给出一个近似比为 $1.5$ 的不同算法。