Single-Sink Network Design with Vertex Connectivity Requirements

We study single-sink network design problems in undirected graphs with vertex connectivity requirements. The input to these problems is an edge-weighted undirected graph $G=(V,E)$, a sink/root vertex $r$, a set of terminals $T \subseteq V$, and integer $k$. The goal is to connect each terminal $t \in T$ to $r$ via $k$ \emph{vertex-disjoint} paths. In the {\em connectivity} problem, the objective is to find a min-cost subgraph of $G$ that contains the desired paths. There is a $2$-approximation for this problem when $k \le 2$ \cite{FleischerJW} but for $k \ge 3$, the first non-trivial approximation was obtained in the recent work of Chakraborty, Chuzhoy and Khanna \cite{ChakCK08}; they describe and analyze an algorithm with an approximation ratio of $O(k^{O(k^2)}\log^4 n)$ where $n=|V|$. In this paper, inspired by the results and ideas in \cite{ChakCK08}, we show an $O(k^{O(k)}\log |T|)$-approximation bound for a simple greedy algorithm. Our analysis is based on the dual of a natural linear program and is of independent technical interest. We use the insights from this analysis to obtain an $O(k^{O(k)}\log |T|)$-approximation for the more general single-sink {\em rent-or-buy} network design problem with vertex connectivity requirements. We further extend the ideas to obtain a poly-logarithmic approximation for the single-sink {\em buy-at-bulk} problem when $k=2$ and the number of cable-types is a fixed constant; we believe that this should extend to any fixed $k$. We also show that for the non-uniform buy-at-bulk problem, for each fixed $k$, a small variant of a simple algorithm suggested by Charikar and Kargiazova \cite{CharikarK05} for the case of $k=1$ gives an $2^{O(\sqrt{\log |T|})}$ approximation for larger $k$. These results show that for each of these problems, simple and natural algorithms that have been developed for $k=1$ have good performance for small $k > 1$.