Approximating Minimum Cost Connectivity Problems via Uncrossable Bifamilies and Spider-Cover Decompositions

• Authors: Zeev Nutov
• Venue: FOCS '09 Proceedings of the 2009 50th Annual IEEE Symposium on Foundations of Computer Science
• Year: 2009

We give approximation algorithms for the {\sf \footnotesize Gene\-ralized Steiner Network} ({\sf \small GSN}) problem. The input consists of a graph $G=(V, E)$ with edge/node costs, a node subset $S \subseteq V$, and connectivity requirements $\{r(s, t):s, t \in T \subseteq V\}$.The goal is to find a minimum cost subgraph $H$ that for all $s, t \in T$contains $r(s, t)$ pairwise edge-disjoint $st$-paths so that no two of them have a node in $S-\{s, t\}$ in common. Three extensively studied particular cases are: {\sf \footnotesize Edge-GSN} ($S=\emptyset$), {\sf \footnotesize Node-GSN} ($S=V$), and {\sf \footnotesize Element-GSN} ($r(s, t)=0$ whenever $s \in S$ or $t \in S$).Let $k=\max_{s, t \in T} r(s, t)$.In {\sf \footnotesize Rooted GSN} there is $s \in T$ so that $r(u, t)=0$ for all $u \neq s$, and in the {\sf \footnotesize Subset {\normalsize $k$}-Connected Subgraph} problem $r(s, t)=k$ for all $s, t \in T$. For edge costs, our ratios are $O(k^2)$ for {\sf \footnotesize Rooted GSN} and $O(k^2 \log k)$ for {\sf \footnotesize Subset {\normalsize $k$}-Connected Subgraph}.This improves the previous ratio $O(k^2 \log n)$ and settles the approximability of these problems to a constant for bounded $k$. For node-cost, our ratios are: \begin{itemize}\item$O(k \log |T|)$ \ for {\sf \footnotesize Element-GSN}, matching the best known ratio for {\sf \footnotesize Edge-GSN}. \item$O(k^2 \log |T|)$ for {\sf \footnotesize Rooted GSN} and $O(k^3 \log |T|)$ for {\sf \footnotesize Subset {\normalsize $k$}-Connected Subgraph}, improving the ratio $O(k^8 \log^2 |T|)$. \item$O(k^4 \log^2 |T|)$ for {\sf \footnotesize GSN}; this is the first non-trivial approximation algorithm for the problem.\end{itemize}