$k$ disjoint $st$-paths activation in polynomial time

TL;DR

This paper presents a polynomial time algorithm for the activation of $k$ disjoint $st$-paths in undirected graphs with power activation costs, solving an open problem for the case $k=2$ and improving related approximation algorithms.

Contribution

It introduces the first polynomial time algorithm for the activation $k$ disjoint $st$-paths problem with power costs, resolving an open question for $k=2$.

Findings

Polynomial time algorithm for activation 2 disjoint $st$-paths.

Implications for activation 2 edge disjoint paths.

Enhanced approximation ratios for min-power $k$-connected subgraph.

Abstract

In activation network design problems we are given an undirected graph $G=(V,E)$ and a pair of activation costs $\{c_e^u,c_e^v\}$ for each $e=uv \in E$. The goal is to find an edge set $F \subseteq E$ that satisfies a prescribed property of minimum activation cost $\tau(F)=\sum_{v \in V} \max \{c_e^v: e \in F \mbox{ is incident to } v\}$. In the Activation $k$ Disjoint Paths problem we are given $s,t \in V$ and an integer $k$, and seek an edge set $F \subseteq E$ of $k$ internally disjoint $st$-paths of minimum activation cost. The problem admits an easy $2$-approximation algorithm. However, it was an open question whether the problem is in P even for $k=2$ and power activation costs, when $c_e^u=c_e^v$ for all $e=uv \in E$. Here we will answer this question by giving a polynomial time algorithm using linear programing. We will also mention several consequences, among them a polynomial time algorithm for the Activation 2 Edge Disjoint Paths problem, and improved approximation ratios for the Min-Power $k$-Connected Subgraph problem.