Algorithms for Single-Source Vertex Connectivity

In the survivable network design problem (SNDP) the goal is to find a minimum cost subset of edges that satisfies a given set of pairwise connectivity requirements among the vertices. This general network design framework has been studied extensively and is tied to the development of major algorithmic techniques. For the edge-connectivity version of the problem, a 2-approximation algorithm is known for arbitrary pairwise connectivity requirements. However, no non-trivial algorithms are known for its vertex connectivity counterpart. In fact, even highly restricted special cases of the vertex connectivity version remain poorly understood.We study the single-source k-vertex connectivity version of SNDP. We are given a graph G(V,E) with a subset T of terminals and a source vertex s, and the goal is to find a minimum cost subset of edges ensuring that every terminal is k-vertex connected to s. Our main result is an O(k log n)-approximation algorithm for this problem; this improves upon the recent 2
<sup>O(k

2

)</sup>
log
⁴
n-approximation. Our algorithm is based on an intuitive rerouting scheme. The analysis relies on a structural result that may be of independent interest: we show that any solution can be decomposed into a disjoint collection of multiple-legged spiders, which are then used to re-route flow from terminals to the source via other terminals.We also obtain the first non-trivial approximation algorithm for the vertex-cost version of the same problem, achieving an O(k
⁷
log
²
n)-approximation.