Approximating {0,1,2}-Survivable Networks with Minimum Number of Steiner Points

TL;DR

This paper introduces improved approximation algorithms for survivable network design problems with minimum Steiner points, achieving better ratios in Euclidean and general metric spaces, especially for low connectivity variants.

Contribution

It presents a new approximation ratio for the Steiner Tree with Minimum Number of Steiner Points problem and a simple  approximation algorithm for low connectivity survivable networks.

Findings

Approximation ratio for ST-MSP in Euclidean space is less than 2.3863.

Introduces a  approximation algorithm for SN-MSP with r;{0,1,2}.

Improves previous ratios for Steiner Forest with Minimum Number of Steiner Points.

Abstract

We consider low connectivity variants of the Survivable Network with Minimum Number of Steiner Points (SN-MSP) problem: given a finite set $R$ of terminals in a metric space (M,d), a subset $B \subseteq R$ of "unstable" terminals, and connectivity requirements {r_{uv}: u,v \in R}, find a minimum size set $S \subseteq M$ of additional points such that the unit-disc graph of $R \cup S$ contains $r_{uv}$ pairwise internally edge-disjoint and $(B \cup S)$-disjoint $uv$-paths for all $u,v \in R$. The case when $r_{uv}=1$ for all $u,v \in R$ is the {\sf Steiner Tree with Minimum Number of Steiner Points} (ST-MSP) problem, and the case $r_{uv} \in \{0,1\}$ is the {\sf Steiner Forest with Minimum Number of Steiner Points} (SF-MSP) problem. Let $\Delta$ be the maximum number of points in a unit ball such that the distance between any two of them is larger than 1. It is known that $\Delta=5$ in $\mathbb{R}^2$ The previous known approximation ratio for {\sf ST-MSP} was $\lfloor (\Delta+1)/2 \rfloor+1+\epsilon$ in an arbitrary normed space \cite{NY}, and $2.5+\epsilon$ in the Euclidean space $\mathbb{R}^2$ \cite{cheng2008relay}. Our approximation ratio for ST-MSP is $1+\ln(\Delta-1)+\epsilon$ in an arbitrary normed space, which in $\mathbb{R}^2$ reduces to $1+\ln 4+\epsilon < 2.3863 +\epsilon$. For SN-MSP with $r_{uv} \in \{0,1,2\}$, we give a simple $\Delta$-approximation algorithm. In particular, for SF-MSP, this improves the previous ratio $2\Delta$.