Additive Approximation for Bounded Degree Survivable Network Design

Abstract

In the minimum bounded degree Steiner network problem, we are given an undirected graph with an edge cost for each edge, a connectivity requirement $r_{uv}$ for each pair of vertices $u$ and $v$, and a degree upper bound $b_v$ for each vertex $v$. The task is to find a minimum cost subgraph that satisfies all the connectivity requirements and degree upper bounds. Let $r_{\max}:=\max_{u,v} \{r_{uv}\}$ and ${\sc opt}$ be the cost of an optimal solution that satisfies all the degree bounds. We present approximation algorithms that minimize the total cost and the degree violation simultaneously. In the special case when $r_{\max}=1$, there is a polynomial time algorithm that returns a Steiner forest of cost at most $2{\sc opt}$ and the degree of each vertex $v$ is at most $b_v+3$. In the general case, there is a polynomial time algorithm that returns a Steiner network of cost at most $2{\sc opt}$ and the degree of each vertex $v$ is at most $b_v+6r_{\max}+3$. The algorithms are based on the iterative relaxation method, and the analysis of the algorithms is nearly tight.